diff --git a/.gitignore b/.gitignore
index ee19a2f..addb5db 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,5 +1,6 @@
Manifest.toml
local/*
+profiles/*
docs/build/*
.vscode/*
.github/*
diff --git a/README.md b/README.md
index 8dc658d..119ff1b 100644
--- a/README.md
+++ b/README.md
@@ -10,7 +10,7 @@ Very early in the development cycle, version 0.2.0.
## Julia implementations of integer triangles.
We give a framework for computing mathematical integer triangles and use
-it to create "Integer Triangle Trait Cards"™.
+it to create so called "Integer Triangle Trait Cards".
A trait card is a compilation of the essential characteristics of an integer triangle,
whereby we understand the characteristics of a triangle to be integer sequences that
@@ -19,8 +19,7 @@ can be obtained from the triangle by elementary transformations.
To see what you can expect start by executing
using IntegerTriangles
- dim = 8
- TraitCard(BinomialTriangle, BinomialTransform, dim)
+ TraitCard(BinomialTriangle, 8)
Overview tables can be automatically generated for a variety of triangles and traits.
@@ -35,10 +34,11 @@ Overview tables can be automatically generated for a variety of triangles and tr
| nothing | Laguerre | Rev | TransNat1 | 1, 3, 15, 97, 753, 6771, 68983, 783945 |
-Note that we assume all sequences to start at offset = 0. Also note that all A-numbers
-are approximativ only, i.e. the first few terms may differ.
+Important: Note that we assume all sequences to start at offset = 0. Also note that all
+references to A-numbers are approximativ only, i.e. the first few terms of the sequence
+may differ and the OEIS-'offset' is always disregarded.
-To use this feature you have to download the file [stripped.gz]( http://oeis.org/stripped.gz) from oeis.org, expand it and put it in the directory ../data.
+To use this feature you have to download the file [stripped.gz](http://oeis.org/stripped.gz) from oeis.org, expand it and put it in the directory ../data.
You can also look at the demo [notebook](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/demos/IntegerTriangles.ipynb).
diff --git a/data/0,0,1,8,64,5.json b/data/0,0,1,8,64,5.json
new file mode 100644
index 0000000..519adf5
--- /dev/null
+++ b/data/0,0,1,8,64,5.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "0,0,1,8,64,540,4920,48720,524160,6108480,76809600,1037836800,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/0,1,5,26,160.json b/data/0,1,5,26,160.json
new file mode 100644
index 0000000..ed9656e
--- /dev/null
+++ b/data/0,1,5,26,160.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "0,1,5,26,160,1140,9240,84000,846720,9374400,113097600,1476921600,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/0,1,6,26,100.json b/data/0,1,6,26,100.json
new file mode 100644
index 0000000..868a26e
--- /dev/null
+++ b/data/0,1,6,26,100.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "0,1,6,26,100,361,1254,4245,14108,46247,149998,482412,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,0,0,1,0,1,.json b/data/1,0,0,1,0,1,.json
new file mode 100644
index 0000000..300e32d
--- /dev/null
+++ b/data/1,0,0,1,0,1,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,0,0,1,0,1,0,1,1,0,1,4,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,0,1,0,1,1,.json b/data/1,0,1,0,1,1,.json
new file mode 100644
index 0000000..caa0239
--- /dev/null
+++ b/data/1,0,1,0,1,1,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,0,1,0,1,1,0,3,4,1,0,23,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,0,1,1,2,5,.json b/data/1,0,1,1,2,5,.json
new file mode 100644
index 0000000..f920c5c
--- /dev/null
+++ b/data/1,0,1,1,2,5,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,0,1,1,2,5,13,38,125,449,1742,7269,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,0,2,0,5,0,.json b/data/1,0,2,0,5,0,.json
new file mode 100644
index 0000000..52214ce
--- /dev/null
+++ b/data/1,0,2,0,5,0,.json
@@ -0,0 +1,352 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,0,2,0,5,0,14,0,42,0,132,0,",
+ "count": 5,
+ "start": 0,
+ "results": [
+ {
+ "number": 126120,
+ "data": "1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,0,1767263190,0,6564120420,0,24466267020,0,91482563640,0",
+ "name": "Catalan numbers (A000108) interpolated with 0's.",
+ "comment": [
+ "Inverse binomial transform of A001006.",
+ "The Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...].",
+ "Counts returning walks of length n on a 1-d integer lattice with step set {+1,-1} which stay in the chamber x \u003e= 0. - _Andrew V. Sutherland_, Feb 29 2008",
+ "Moment sequence of the trace of a random matrix in G=USp(2)=SU(2). If X=tr(A) is a random variable (A distributed according to the Haar measure on G) then a(n) = E[X^n]. - _Andrew V. Sutherland_, Feb 29 2008",
+ "Essentially the same as A097331. - _R. J. Mathar_, Jun 15 2008",
+ "Number of distinct proper binary trees with n nodes. - Chris R. Sims (chris.r.sims(AT)gmail.com), Jun 30 2010",
+ "Number of n-step walks that start and end at origin with the constraint that they are never negative. - _Benjamin Phillabaum_, Mar 07 2011",
+ "-a(n-1), with a(-1):=0, n\u003e=0, is the Z-sequence for the Riordan array A049310 (Chebyshev S). For the definition see that triangle. - _Wolfdieter Lang_, Nov 04 2011",
+ "See A180874 (also A238390 and A097610) and A263916 for relations to the general Bell A036040, cycle index A036039, and cumulant expansion polynomials A127671 through the Faber polynomials. - _Tom Copeland_, Jan 26 2016",
+ "Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-1,1}. - _David Nguyen_, Dec 20 2016",
+ "A signed version is generated by evaluating polynomials in A126216 that are essentially the face polynomials of the associahedra. This entry's sequence is related to an inversion relation on p. 34 of Mizera, related to Feynman diagrams. - _Tom Copeland_, Dec 09 2019"
+ ],
+ "reference": [
+ "Jerome Spanier and Keith B. Oldham, \"Atlas of Functions\", Ch. 49, Hemisphere Publishing Corp., 1987."
+ ],
+ "link": [
+ "G. C. Greubel, \u003ca href=\"/A126120/b126120.txt\"\u003eTable of n, a(n) for n = 0..1000\u003c/a\u003e",
+ "V. E. Adler, \u003ca href=\"http://arxiv.org/abs/1510.02900\"\u003eSet partitions and integrable hierarchies\u003c/a\u003e, arXiv:1510.02900 [nlin.SI], 2015.",
+ "Martin Aigner, \u003ca href=\"http://dx.doi.org/10.1007/978-88-470-2107-5_15\"\u003eCatalan and other numbers: a recurrent theme\u003c/a\u003e, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.",
+ "Andrei Asinowski, Cyril Banderier, and Valerie Roitner, \u003ca href=\"https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf\"\u003eGenerating functions for lattice paths with several forbidden patterns\u003c/a\u003e, (2019).",
+ "C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, \u003ca href=\"https://arxiv.org/abs/1609.06473\"\u003eExplicit formulas for enumeration of lattice paths: basketball and the kernel method\u003c/a\u003e, arXiv:1609.06473 [math.CO], 2016.",
+ "Radica Bojicic, Marko D. Petkovic and Paul Barry, \u003ca href=\"http://arxiv.org/abs/1112.1656\"\u003eHankel transform of a sequence obtained by series reversion II-aerating transforms\u003c/a\u003e, arXiv:1112.1656 [math.CO], 2011.",
+ "Colin Defant, \u003ca href=\"https://arxiv.org/abs/2004.11367\"\u003eTroupes, Cumulants, and Stack-Sorting\u003c/a\u003e, arXiv:2004.11367 [math.CO], 2020.",
+ "Isaac DeJager, Madeleine Naquin, Frank Seidl, \u003ca href=\"https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf\"\u003eColored Motzkin Paths of Higher Order\u003c/a\u003e, VERUM 2019.",
+ "Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, \u003ca href=\"http://arxiv.org/abs/1110.6638\"\u003eSato-Tate distributions and Galois endomorphism modules in genus 2\u003c/a\u003e, arXiv:1110.6638 [math.NT], 2011.",
+ "Aoife Hennessy, \u003ca href=\"http://repository.wit.ie/1693/1/AoifeThesis.pdf\"\u003eA Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths\u003c/a\u003e, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.",
+ "Kiran S. Kedlaya and Andrew V. Sutherland, \u003ca href=\"http://dspace.mit.edu/handle/1721.1/64701\"\u003eHyperellipticCurves, L-Polynomials, and Random Matrices\u003c/a\u003e. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)",
+ "S. Mizera, \u003ca href=\"https://arxiv.org/abs/1706.08527\"\u003eCombinatorics and Topology of Kawai-Lewellen-Tye Relations\u003c/a\u003e, arXiv:1706.08527 [hep-th], 2017.",
+ "E. Rowland, \u003ca href=\"http://dx.doi.org/10.1016/j.jcta.201.03.004\"\u003ePattern avoidance in binary trees\u003c/a\u003e, J. Comb. Theory A 117 (6) (2010) 741-758, Sec. 3.1.",
+ "Yidong Sun and Fei Ma, \u003ca href=\"http://arxiv.org/abs/1305.2015\"\u003eMinors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths\u003c/a\u003e, arXiv:1305.2015 [math.CO], 2013.",
+ "Y. Wang and Z.-H. Zhang, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL18/Wang/wang21.html\"\u003eCombinatorics of Generalized Motzkin Numbers\u003c/a\u003e, J. Int. Seq. 18 (2015) # 15.2.4."
+ ],
+ "formula": [
+ "a(2*n) = A000108(n), a(2*n+1) = 0.",
+ "a(n) = A053121(n,0).",
+ "(1/Pi) Integral_{0 .. Pi} (2*cos(x))^n *2*sin^2(x) dx. - _Andrew V. Sutherland_, Feb 29 2008",
+ "G.f.: 1/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-...(continued fraction). - _Philippe Deléham_, Nov 24 2009",
+ "G.f. A(x) satisfies A(x) = 1 + x^2*A(x)^2. - _Vladimir Kruchinin_, Feb 18 2011",
+ "E.g.f.: I_1(2x)/x Where I_n(x) is the modified Bessel function. - _Benjamin Phillabaum_, Mar 07 2011",
+ "Apart from the first term the e.g.f. is given by x*HyperGeom([1/2],[3/2,2], x^2). - _Benjamin Phillabaum_, Mar 07 2011",
+ "a(n) = Integral_{x=-2..2} x^n*sqrt((2-x)*(2+x)))/(2*Pi). - _Peter Luschny_, Sep 11 2011",
+ "E.g.f.: E(0)/(1-x) where E(k) = 1-x/(1-x/(x-(k+1)*(k+2)/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 05 2013",
+ "G.f.: 3/2- sqrt(1-4*x^2)/2 = 1/x^2 + R(0)/x^2, where R(k) = 2*k-1 - x^2*(2*k-1)*(2*k+1)/R(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 28 2013",
+ "G.f.: 1/Q(0), where Q(k) = 2*k+1 + x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 09 2014",
+ "a(n) = n!*[x^n]hypergeom([],[2],x^2). - _Peter Luschny_, Jan 31 2015",
+ "a(n) = 2^n*hypergeom([3/2,-n],[3],2). - _Peter Luschny_, Feb 03 2015",
+ "a(n) = ((-1)^n+1)*2^(2*floor(n/2)-1)*Gamma(floor(n/2)+1/2)/(sqrt(Pi)* Gamma(floor(n/2)+2)). - _Ilya Gutkovskiy_, Jul 23 2016",
+ "D-finite with recurrence (n+2)*a(n) +4*(-n+1)*a(n-2)=0. - _R. J. Mathar_, Mar 21 2021"
+ ],
+ "example": [
+ "G.f. = 1 + x^2 + 2*x^4 + 5*x^6 + 14*x^8 + 42*x^10 + 132*x^12 + 429*x^14 + ..."
+ ],
+ "maple": [
+ "with(combstruct): grammar := { BB = Sequence(Prod(a,BB,b)), a = Atom, b = Atom }: seq(count([BB,grammar], size=n),n=0..47); # _Zerinvary Lajos_, Apr 25 2007",
+ "BB := {E=Prod(Z,Z), S=Union(Epsilon,Prod(S,S,E))}: ZL:=[S,BB,unlabeled]: seq(count(ZL, size=n), n=0..45); # _Zerinvary Lajos_, Apr 22 2007",
+ "BB := [T,{T=Prod(Z,Z,Z,F,F), F=Sequence(B), B=Prod(F,Z,Z)}, unlabeled]: seq(count(BB, size=n+1), n=0..45); # valid for n\u003e 0. # _Zerinvary Lajos_, Apr 22 2007",
+ "seq(n!*coeff(series(hypergeom([],[2],x^2),x,n+2),x,n),n=0..45); # _Peter Luschny_, Jan 31 2015"
+ ],
+ "mathematica": [
+ "a[n_?EvenQ] := CatalanNumber[n/2]; a[n_] = 0; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Sep 10 2012 *)",
+ "a[ n_] := If[ n \u003c 0, 0, n! SeriesCoefficient[ BesselI[ 1, 2 x] / x, {x, 0, n}]]; (* _Michael Somos_, Mar 19 2014 *)"
+ ],
+ "program": [
+ "(Sage)",
+ "def A126120_list(n) :",
+ " D = [0]*(n+2); D[1] = 1",
+ " b = True; h = 2; R = []",
+ " for i in range(2*n-1) :",
+ " if b :",
+ " for k in range(h,0,-1) : D[k] -= D[k-1]",
+ " h += 1; R.append(abs(D[1]))",
+ " else :",
+ " for k in range(1,h, 1) : D[k] += D[k+1]",
+ " b = not b",
+ " return R",
+ "A126120_list(46) # _Peter Luschny_, Jun 03 2012",
+ "(MAGMA) \u0026cat [[Catalan(n), 0]: n in [0..30]]; // _Vincenzo Librandi_, Jul 28 2016"
+ ],
+ "xref": [
+ "Cf. A000108.",
+ "Cf. A036039, A036040, A097610, A127671, A180874, A238390, A263916.",
+ "Cf. A126216."
+ ],
+ "keyword": "nonn",
+ "offset": "0,5",
+ "author": "_Philippe Deléham_, Mar 06 2007",
+ "ext": [
+ "An erroneous comment removed by _Tom Copeland_, Jul 23 2016"
+ ],
+ "references": 48,
+ "revision": 181,
+ "time": "2021-03-21T10:15:38-04:00",
+ "created": "2007-05-11T03:00:00-04:00"
+ },
+ {
+ "number": 97331,
+ "data": "1,1,0,1,0,2,0,5,0,14,0,42,0,132,0,429,0,1430,0,4862,0,16796,0,58786,0,208012,0,742900,0,2674440,0,9694845,0,35357670,0,129644790,0,477638700,0,1767263190,0,6564120420,0,24466267020,0,91482563640,0,343059613650,0",
+ "name": "Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).",
+ "comment": [
+ "Binomial transform is A097332. Second binomial transform is A014318.",
+ "Essentially the same as A126120. - _R. J. Mathar_, Jun 15 2008",
+ "Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - _Paul Barry_, Aug 10 2009"
+ ],
+ "link": [
+ "Michael De Vlieger, \u003ca href=\"/A097331/b097331.txt\"\u003eTable of n, a(n) for n = 0..3340\u003c/a\u003e",
+ "Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, \u003ca href=\"http://math.colgate.edu/~integers/t46/t46.Abstract.html\"\u003eMotzkin paths with a restricted first return decomposition\u003c/a\u003e, Integers (2019) Vol. 19, A46."
+ ],
+ "formula": [
+ "a(n) = 0^n + Catalan((n-1)/2)(1-(-1)^n)/2.",
+ "Unsigned version of A090192, A105523. - _Philippe Deléham_, Sep 29 2006",
+ "From _Paul Barry_, Aug 10 2009: (Start)",
+ "G.f.: 1+xc(x^2), c(x) the g.f. of A000108;",
+ "G.f.: 1/(1-x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+... (continued fraction);",
+ "G.f.: 1+x/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). (End)",
+ "G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+2*x) (continued fraction); more generally g.f. C(x/(1+2*x)) where C(x) is the g.f. for the Catalan numbers (A000108). - _Joerg Arndt_, Mar 18 2011",
+ "Conjecture: (n+1)*a(n) + n*a(n-1) + 4*(-n+2)*a(n-2) + 4*(-n+3)*a(n-3)=0. - _R. J. Mathar_, Dec 02 2012",
+ "Recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ 2^(n+1)/((n+1)^(3/2)*sqrt(2*Pi)). - _Fung Lam_, Mar 17 2014"
+ ],
+ "maple": [
+ "A097331_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;",
+ "for w from 1 to n do a[w]:=a[w-1]-(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A097331_list(48); # _Peter Luschny_, May 19 2011"
+ ],
+ "mathematica": [
+ "a[0] = 1; a[n_?OddQ] := CatalanNumber[(n-1)/2]; a[_] = 0; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover_, Jul 24 2013 *)"
+ ],
+ "program": [
+ "(Sage)",
+ "def A097331_list(n) :",
+ " D = [0]*(n+2); D[1] = 1",
+ " b = True; h = 1; R = []",
+ " for i in range(2*n-1) :",
+ " if b :",
+ " for k in range(h,0,-1) : D[k] -= D[k-1]",
+ " h += 1; R.append(abs(D[1]))",
+ " else :",
+ " for k in range(1,h, 1) : D[k] += D[k+1]",
+ " b = not b",
+ " return R",
+ "A097331_list(49) # _Peter Luschny_, Jun 03 2012"
+ ],
+ "keyword": "easy,nonn",
+ "offset": "0,6",
+ "author": "_Paul Barry_, Aug 05 2004",
+ "references": 12,
+ "revision": 36,
+ "time": "2020-01-28T18:48:36-05:00",
+ "created": "2004-09-22T03:00:00-04:00"
+ },
+ {
+ "number": 90192,
+ "data": "1,1,0,-1,0,2,0,-5,0,14,0,-42,0,132,0,-429,0,1430,0,-4862,0,16796,0,-58786,0,208012,0,-742900,0,2674440,0,-9694845,0,35357670,0,-129644790,0,477638700,0,-1767263190,0,6564120420,0,-24466267020,0,91482563640,0,-343059613650,0",
+ "name": "Carlitz-Riordan q-Catalan numbers (recurrence version) for q = -1.",
+ "comment": [
+ "Hankel transform is (-1)^C(n+1,2). - _Paul Barry_, Feb 15 2008"
+ ],
+ "link": [
+ "Fung Lam and Seiichi Manyama, \u003ca href=\"/A090192/b090192.txt\"\u003eTable of n, a(n) for n = 0..3338\u003c/a\u003e (first 1002 terms from Fung Lam)"
+ ],
+ "formula": [
+ "a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-1 and a(0)=1.",
+ "G.f.: 1+x*c(-x^2), where c(x) is the g.f. of A000108; a(n) = 0^n+C((n-1)/2)(-1)^((n-1)/2)(1-(-1)^n)/2, where C(n) = A000108(n). - _Paul Barry_, Feb 15 2008",
+ "G.f.: 1/(1-x/(1+x/(1-x/(1+x/(1-x/(1+x/(1-.... (continued fraction). - _Paul Barry_, Jan 15 2009",
+ "a(n) = 2 * a(n-1) - Sum_{k=1..n} a(k-1) * a(n-k) if n\u003e0. - _Michael Somos_, Jul 23 2011",
+ "G.f.: (2*x-1+sqrt(1+4*x^2))/(2*x). - _Philippe Deléham_, Nov 07 2011",
+ "E.g.f.: x*hypergeom([1/2],[2,3/2],-x^2)=A(x)=x*(1-(x^2)/(Q(0)+(x^2)); Q(k)=2*(k^3)+9*(k^2)+(13-2*(x^2))*k-(x^2)+6+(x^2)*(k+1)*(k+2)*((2*k+3)^2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 22 2011",
+ "G.f.: 2 + (G(0)-1)/(2*x) where G(k)=1 - 4*x/(1 + 1/G(k+1) ); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 08 2012",
+ "G.f.: 2 + (G(0) -1)/x, where G(k)= 1 - x/(1 + x/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013",
+ "G.f.: 1 - 1/(2*x) + G(0)/(4*x), where G(k)= 1 + 1/(1 - 2*x^2*(2*k-1)/(2*x^2*(2*k-1) - (k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013",
+ "G.f.: 1- x/(Q(0) + 2*x^2), where Q(k)= (4*x^2 - 1)*k - 2*x^2 - 1 + 2*x^2*(k+1)*(2*k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013",
+ "G.f.: 1+ x/Q(0), where Q(k) = 2*k+1 - x^2*(1-4*(k+1)^2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 09 2014",
+ "D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ (-1)^((n+3)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - _Fung Lam_, Mar 17 2014",
+ "a(n) = hypergeom([-n+1,-n], [2], -1). - _Peter Luschny_, Sep 22 2014",
+ "G.f. A(x) satisfies A(x) = 1 / (1 - x * A(-x)). - _Michael Somos_, Dec 26 2016"
+ ],
+ "example": [
+ "G.f. = 1 + x - x^3 + 2*x^5 - 5*x^7 + 14*x^9 - 42*x^11 + 132*x^13 - 429*x^15 + ..."
+ ],
+ "maple": [
+ "A090192_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;",
+ "for w from 1 to n do a[w] := a[w-1]-add(a[j]*a[w-j-1], j=1..w-1) od;",
+ "convert(a, list) end: A090192_list(48); # _Peter Luschny_, May 19 2011",
+ "a := n -\u003e hypergeom([-n+1,-n],[2],-1); seq(round(evalf(a(n), 69)), n=0..48); # _Peter Luschny_, Sep 22 2014",
+ "a:= proc(n) if n::even then 0 else (-1)^((n-1)/2)*binomial(n+1,(n+1)/2)/(2*n) fi end proc: a(0):= 1:",
+ "seq(a(n), n=0..100); # _Robert Israel_, Sep 22 2014"
+ ],
+ "mathematica": [
+ "CoefficientList[Series[(2 x - 1 + Sqrt[1 + 4*x^2])/(2 x), {x, 0, 50}],",
+ " x] (* _G. C. Greubel_, Dec 24 2016 *)",
+ "Table[Hypergeometric2F1[1 - n, -n, 2, -1], {n, 0, 48}] (* _Michael De Vlieger_, Dec 26 2016 *)"
+ ],
+ "program": [
+ "(PARI) {a(n) = my(A); if( n\u003c0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = 2 * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 23 2011 */",
+ "(Sage)",
+ "def A090192_list(n) :",
+ " D = [0]*(n+2); D[1] = 1",
+ " b = True; h = 1; R = []",
+ " for i in range(2*n-1) :",
+ " if b :",
+ " for k in range(h,0,-1) : D[k] -= D[k-1]",
+ " h += 1; R.append(D[1])",
+ " else :",
+ " for k in range(1,h, 1) : D[k] += D[k+1]",
+ " b = not b",
+ " return R",
+ "A090192_list(49) # _Peter Luschny_, Jun 03 2012",
+ "(Ruby)",
+ "def A(q, n)",
+ " ary = [1]",
+ " (1..n).each{|i| ary \u003c\u003c (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}",
+ " ary",
+ "end",
+ "def A090192(n)",
+ " A(-1, n)",
+ "end # _Seiichi Manyama_, Dec 24 2016",
+ "(PARI) Vec((2*x - 1 + sqrt(1+4*x^2))/(2*x) + O(x^50)) \\\\ _G. C. Greubel_, Dec 24 2016"
+ ],
+ "xref": [
+ "Cf. A227543.",
+ "Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), this sequence (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).",
+ "Column k=1 of A290789."
+ ],
+ "keyword": "sign",
+ "offset": "0,6",
+ "author": "_Philippe Deléham_, Jan 22 2004",
+ "references": 33,
+ "revision": 109,
+ "time": "2020-04-10T01:31:10-04:00",
+ "created": "2004-02-19T03:00:00-05:00"
+ },
+ {
+ "number": 105523,
+ "data": "1,-1,0,1,0,-2,0,5,0,-14,0,42,0,-132,0,429,0,-1430,0,4862,0,-16796,0,58786,0,-208012,0,742900,0,-2674440,0,9694845,0,-35357670,0,129644790,0,-477638700,0,1767263190,0",
+ "name": "Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.",
+ "comment": [
+ "Row sums of A105522. Row sums of inverse of A105438.",
+ "First column of number triangle A106180."
+ ],
+ "link": [
+ "Vincenzo Librandi, \u003ca href=\"/A105523/b105523.txt\"\u003eTable of n, a(n) for n = 0..1000\u003c/a\u003e",
+ "R. J. Martin and M. J. Kearney, \u003ca href=\"http://dx.doi.org/10.1007/s00010-010-0051-0\"\u003eAn exactly solvable self-convolutive recurrence\u003c/a\u003e, Aequat. Math., 80 (2010), 291-318. see p. 313.",
+ "R. J. Martin and M. J. Kearney, \u003ca href=\"http://arXiv.org/abs/1103.4936\"\u003eAn exactly solvable self-convolutive recurrence\u003c/a\u003e, arXiv:1103.4936 [math.CO]"
+ ],
+ "formula": [
+ "G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).",
+ "a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).",
+ "G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - _Paul Barry_, Jan 15 2009",
+ "a(n) = Sum{k=0..n, A090181(n,k)*(-1)^k}. - _Philippe Deléham_, Feb 02 2009",
+ "a(n) = (1/n)*sum((-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1), i=0..n-1). - _Vladimir Kruchinin_, Dec 26 2010",
+ "With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n\u003e1. - _Michael Somos_, Jul 25 2011",
+ "D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - _Fung Lam_, Mar 18 2014",
+ "For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - _Fung Lam_, Mar 17 2014",
+ "a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - _Peter Luschny_, Oct 31 2014"
+ ],
+ "example": [
+ "G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ..."
+ ],
+ "maple": [
+ "A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;",
+ "for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A105523_list(40); # _Peter Luschny_, May 19 2011"
+ ],
+ "mathematica": [
+ "a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 14 2011, after _Vladimir Kruchinin_ *)",
+ "CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 01 2014 *)",
+ "a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* _Michael Somos_, Jun 17 2015 *)",
+ "a[ n_] := If[ n \u003c 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* _Michael Somos_, Jun 17 2015 *)"
+ ],
+ "program": [
+ "(PARI) {a(n) = local(A); if( n\u003c0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 24 2011 */",
+ "(Sage)",
+ "def A105523(n):",
+ " if is_even(n): return 0 if n\u003e0 else 1",
+ " return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))",
+ "[A105523(n) for n in (0..29)] # _Peter Luschny_, Oct 31 2014",
+ "(MAGMA) m:=25; R\u003cx\u003e:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // _G. C. Greubel_, Sep 16 2018"
+ ],
+ "xref": [
+ "Cf. A000108, A097331, A090192, A090181, A105522, A105438."
+ ],
+ "keyword": "easy,sign",
+ "offset": "0,6",
+ "author": "_Paul Barry_, Apr 11 2005",
+ "ext": [
+ "Typo in definition corrected by _Robert Israel_, Oct 31 2014"
+ ],
+ "references": 15,
+ "revision": 59,
+ "time": "2020-02-21T06:33:16-05:00",
+ "created": "2005-07-19T03:00:00-04:00"
+ },
+ {
+ "number": 210628,
+ "data": "1,-1,0,-1,0,-2,0,-5,0,-14,0,-42,0,-132,0,-429,0,-1430,0,-4862,0,-16796,0,-58786,0,-208012,0,-742900,0,-2674440,0,-9694845,0,-35357670,0,-129644790,0,-477638700,0,-1767263190,0,-6564120420,0,-24466267020,0",
+ "name": "Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.",
+ "comment": [
+ "Except for the leading term, the sequence is equal to -A097331(n). - _Fung Lam_, Mar 22 2014"
+ ],
+ "link": [
+ "Vincenzo Librandi, \u003ca href=\"/A210628/b210628.txt\"\u003eTable of n, a(n) for n = 0..1000\u003c/a\u003e"
+ ],
+ "formula": [
+ "G.f.: 1 - (2*x) / (1 + sqrt( 1 - 4*x^2)) = 1 - (1 - sqrt( 1 - 4*x^2)) / (2*x).",
+ "G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).",
+ "G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.",
+ "G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).",
+ "G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).",
+ "G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).",
+ "G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).",
+ "G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - _Michael Somos_, Jan 02 2013",
+ "a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).",
+ "Convolution inverse of A210736. - _Michael Somos_, Jan 02 2013",
+ "G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 24 2013",
+ "D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - _Fung Lam_, Mar 17 2014",
+ "For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - _Fung Lam_, Mar 17 2014"
+ ],
+ "example": [
+ "G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ..."
+ ],
+ "mathematica": [
+ "CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* _Bruno Berselli_, Mar 25 2012 *)",
+ "a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];"
+ ],
+ "program": [
+ "(PARI) {a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};",
+ "(PARI) {a(n) = if( n\u003c1, n==0, polcoeff( serreverse( -x / (1 + x^2) + x * O(x^n)), n))};",
+ "(PARI) {a(n) = my(A); if( n\u003c0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};",
+ "(Maxima) makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); \\\\ _Bruno Berselli_, Mar 25 2012",
+ "(MAGMA) m:=50; R\u003cx\u003e:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // _G. C. Greubel_, Aug 11 2018"
+ ],
+ "xref": [
+ "Cf. A000108, A001405, A090192, A097331, A126930, A210736."
+ ],
+ "keyword": "sign",
+ "offset": "0,6",
+ "author": "_Michael Somos_, Mar 25 2012",
+ "references": 2,
+ "revision": 56,
+ "time": "2020-01-30T21:29:16-05:00",
+ "created": "2012-03-26T00:35:04-04:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,1,0,1,1,0,.json b/data/1,1,0,1,1,0,.json
new file mode 100644
index 0000000..dd1986e
--- /dev/null
+++ b/data/1,1,0,1,1,0,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,0,1,1,0,1,4,3,0,1,11,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,1,0,1,2,0,.json b/data/1,1,0,1,2,0,.json
new file mode 100644
index 0000000..fc8cfe1
--- /dev/null
+++ b/data/1,1,0,1,2,0,.json
@@ -0,0 +1,87 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,0,1,2,0,1,4,6,0,1,6,",
+ "count": 2,
+ "start": 0,
+ "results": [
+ {
+ "number": 287318,
+ "data": "1,1,0,1,2,0,1,4,6,0,1,6,36,20,0,1,8,90,400,70,0,1,10,168,1860,4900,252,0,1,12,270,5120,44730,63504,924,0,1,14,396,10900,190120,1172556,853776,3432,0,1,16,546,19920,551950,7939008,32496156,11778624,12870,0",
+ "name": "Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.",
+ "formula": [
+ "A(n,k) = A287316(n,k) * binomial(2*n,n)."
+ ],
+ "example": [
+ "Arrays start:",
+ "k\\n| 0 1 2 3 4 5 6",
+ "---|---------------------------------------------------------",
+ "k=0| 1, 0, 0, 0, 0, 0, 0, ... A000007",
+ "k=1| 1, 2, 6, 20, 70, 252, 924, ... A000984",
+ "k=2| 1, 4, 36, 400, 4900, 63504, 853776, ... A002894",
+ "k=3| 1, 6, 90, 1860, 44730, 1172556, 32496156, ... A002896",
+ "k=4| 1, 8, 168, 5120, 190120, 7939008, 357713664, ... A039699",
+ "k=5| 1, 10, 270, 10900, 551950, 32232060, 2070891900, ... A287317",
+ "k=6| 1, 12, 396, 19920, 1281420, 96807312, 8175770064, ...",
+ "k=7| 1, 14, 546, 32900, 2570050, 238935564, 25142196156, ...",
+ "k=8| 1, 16, 720, 50560, 4649680, 514031616, 64941883776, ...",
+ "k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ..."
+ ],
+ "maple": [
+ "A287318_row := proc(k, len) local b, ser;",
+ "b := k -\u003e BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);",
+ "seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:",
+ "for k from 0 to 6 do A287318_row(k, 9) od;"
+ ],
+ "mathematica": [
+ "Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]"
+ ],
+ "xref": [
+ "Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5).",
+ "Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).",
+ "Main diagonal gives A303503.",
+ "Cf. A287316."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "0,5",
+ "author": "_Peter Luschny_, May 23 2017",
+ "references": 3,
+ "revision": 19,
+ "time": "2018-05-02T11:50:20-04:00",
+ "created": "2017-05-23T09:39:37-04:00"
+ },
+ {
+ "number": 329020,
+ "data": "1,1,0,1,2,0,1,4,6,0,1,6,44,20,0,1,8,146,580,70,0,1,10,344,4332,8092,252,0,1,12,670,18152,135954,116304,924,0,1,14,1156,55252,1012664,4395456,1703636,3432,0,1,16,1834,137292,4816030,58199208,144840476,25288120,12870,0",
+ "name": "Square array T(n,k), n\u003e=0, k\u003e=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( Sum_{j=1..k} x_j^(2*j-1) + x_j^(-(2*j-1)) )^(2*n).",
+ "link": [
+ "Seiichi Manyama, \u003ca href=\"/A329020/b329020.txt\"\u003eAntidiagonals n = 0..50, flattened\u003c/a\u003e"
+ ],
+ "formula": [
+ "T(n,k) = Sum_{j=0..floor((2*k-1)*n/(2*k))} (-1)^j * binomial(2*n,j) * binomial((2*k+1)*n-2*k*j-1,(2*k-1)*n-2*k*j) for k \u003e 0."
+ ],
+ "example": [
+ "(x^3 + x + 1/x + 1/x^3)^2 = x^6 + 2*x^4 + 3*x^2 + 4 + 3/x^2 + 2/x^4 + 1/x^6. So T(1,2) = 4.",
+ "Square array begins:",
+ " 1, 1, 1, 1, 1, 1, ...",
+ " 0, 2, 4, 6, 8, 10, ...",
+ " 0, 6, 44, 146, 344, 670, ...",
+ " 0, 20, 580, 4332, 18152, 55252, ...",
+ " 0, 70, 8092, 135954, 1012664, 4816030, ...",
+ " 0, 252, 116304, 4395456, 58199208, 432457640, ..."
+ ],
+ "xref": [
+ "Columns k=0-3 give A000007, A000984, A005721, A063419.",
+ "Rows n=0-2 give A000012, A005843, 2*A143166.",
+ "Main diagonal gives A329021.",
+ "Cf. A077042."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "0,5",
+ "author": "_Seiichi Manyama_, Nov 02 2019",
+ "references": 2,
+ "revision": 34,
+ "time": "2019-11-04T02:21:15-05:00",
+ "created": "2019-11-02T20:00:59-04:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,1,1,0,1,1,.json b/data/1,1,1,0,1,1,.json
new file mode 100644
index 0000000..9b64933
--- /dev/null
+++ b/data/1,1,1,0,1,1,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,1,0,1,1,1,4,0,1,11,1,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,1,1,1,2,0,.json b/data/1,1,1,1,2,0,.json
new file mode 100644
index 0000000..c2ff6f5
--- /dev/null
+++ b/data/1,1,1,1,2,0,.json
@@ -0,0 +1,88 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,1,1,2,0,1,3,1,1,1,4,",
+ "count": 3,
+ "start": 0,
+ "results": [
+ {
+ "number": 78805,
+ "data": "1,1,1,1,2,0,1,3,1,1,1,4,3,2,0,1,5,6,4,2,1,1,6,10,8,6,2,0,1,7,15,15,13,6,3,1,1,8,21,26,25,16,9,2,0,1,9,28,42,45,36,22,9,4,1,1,10,36,64,77,72,50,28,12,2,0,1,11,45,93,126,133,106,70,34,13,5,1,1,12,55,130,198,232",
+ "name": "Triangular array T given by T(n,k)= number of 01-words of length n having exactly k 1's, every runlength of 1's odd and initial letter 0.",
+ "comment": [
+ "Row sums: A028495."
+ ],
+ "reference": [
+ "Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151."
+ ],
+ "formula": [
+ "T(n, k)=T(n-2, k)+T(n-2, k-1)+T(n-2, k-2)+T(n-3, k-1)-T(n-4, k-2) for 0\u003c=k\u003c=n, n\u003e=1. (All numbers T(i, j) not in the array are 0, by definition of T.)"
+ ],
+ "example": [
+ "T(5,2) counts the words 01010, 01001, 00101. Top of triangle T:",
+ "1 = T(1,0)",
+ "1 1 = T(2,0) T(2,1)",
+ "1 2 0 = T(3,0) T(3,1) T(3,2)",
+ "1 3 1 1",
+ "1 4 3 2 0"
+ ],
+ "xref": [
+ "Cf. A078804, A078806."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "1,5",
+ "author": "_Clark Kimberling_, Dec 07 2002",
+ "references": 2,
+ "revision": 6,
+ "time": "2012-03-30T18:57:05-04:00",
+ "created": "2003-05-16T03:00:00-04:00"
+ },
+ {
+ "number": 65432,
+ "data": "1,1,-1,1,-2,0,1,-3,1,1,1,-4,3,2,0,1,-5,6,2,-2,-2,1,-6,10,0,-6,-4,0,1,-7,15,-5,-11,-3,5,5,1,-8,21,-14,-15,4,15,10,0,1,-9,28,-28,-15,19,26,6,-14,-14,1,-10,36,-48,-7,42,30,-16,-42,-28,0,1,-11,45,-75,14,70,16,-60,-70,-14,42,42,1,-12,55,-110,54,96,-28,-120",
+ "name": "Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).",
+ "comment": [
+ "Sums of odd rows are 0, of even rows are the Catalan numbers (A000108) with alternating signs. Row sums of unsigned version give A065441."
+ ],
+ "formula": [
+ "a[0, 0] := 1; a[n_, k_] := 0/;(k \u003e n||n \u003c 0||k \u003c 0); a[n_, k_] := a[n, k] = a[n, k-1]-2a[n-1, k-1]+a[n-1, k]; Table[a[n, k], {n, 0, 16}, {k, 0, n}]"
+ ],
+ "example": [
+ "{1},{1,-1},{1,-2,0},{1,-3,1,1},{1,-4,3,2,0}"
+ ],
+ "keyword": "sign,tabl",
+ "offset": "0,5",
+ "author": "_Wouter Meeussen_, Nov 16 2001",
+ "references": 2,
+ "revision": 6,
+ "time": "2016-04-25T13:17:22-04:00",
+ "created": "2003-05-16T03:00:00-04:00"
+ },
+ {
+ "number": 94184,
+ "data": "1,1,1,1,2,0,1,3,1,-1,1,4,3,-2,0,1,5,6,-2,-2,2,1,6,10,0,-6,4,0,1,7,15,5,-11,3,5,-5,1,8,21,14,-15,-4,15,-10,0,1,9,28,28,-15,-19,26,-6,-14,14,1,10,36,48,-7,-42,30,16,-42,28,0,1,11,45,75,14,-70,16,60,-70,14,42,-42,1,12,55,110,54,-96,-28,120,-70,-56,126,-84,0,1",
+ "name": "Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.",
+ "comment": [
+ "Row sums are A086990 or A090412. (Superseeker finds that the j-th coefficient of OGF(A090412)(z)*(1-z)^j equals A049122). Same absolute values as A065432. Even rows end in 0, odd rows end in Catalan numbers (A000118) with alternating sign."
+ ],
+ "formula": [
+ "T(i, j)=T(i-1, j)-T(i, j-1)+2*T(i-1, j-1), with T(i, 0)=1 and T(i, j)=0 if j\u003ei."
+ ],
+ "example": [
+ "Table starts {1},{1,1},{1,2,0},{1,3,1,-1},{1,4,3,-2,0},{1,5,6,-2,-2,2}"
+ ],
+ "mathematica": [
+ "T[_, 0]:=1;T[0, 0]:=1;T[i_, j_]/;j\u003ei:=0;T[i_, j_]:=T[i, j]=T[i-1, j]-T[i, j-1]+2 T[i-1, j-1]"
+ ],
+ "xref": [
+ "Cf. A086990, A090412, A049122, A009766, A065432, A065441."
+ ],
+ "keyword": "sign,tabl",
+ "offset": "0,5",
+ "author": "_Wouter Meeussen_, May 06 2004",
+ "references": 0,
+ "revision": 4,
+ "time": "2012-03-30T18:37:44-04:00",
+ "created": "2004-06-12T03:00:00-04:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,1,1,1,2,1,.json b/data/1,1,1,1,2,1,.json
new file mode 100644
index 0000000..b3a2161
--- /dev/null
+++ b/data/1,1,1,1,2,1,.json
@@ -0,0 +1,591 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,1,1,2,1,1,3,2,1,1,4,",
+ "count": 16,
+ "start": 0,
+ "results": [
+ {
+ "number": 77592,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,3,3,1,1,5,4,6,2,1,1,6,5,10,3,4,1,1,7,6,15,4,9,2,1,1,8,7,21,5,16,3,4,1,1,9,8,28,6,25,4,10,3,1,1,10,9,36,7,36,5,20,6,4,1,1,11,10,45,8,49,6,35,10,9,2,1,1,12,11,55,9,64,7,56,15,16,3,6,1",
+ "name": "Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.",
+ "link": [
+ "Alois P. Heinz, \u003ca href=\"/A077592/b077592.txt\"\u003eAntidiagonals n = 1..141, flattened\u003c/a\u003e",
+ "Adolf Piltz, \u003ca href=\"https://gdz.sub.uni-goettingen.de/id/PPN271032898\"\u003eUeber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst\u003c/a\u003e, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.",
+ "Wikipedia, \u003ca href=\"https://de.wikipedia.org/wiki/Adolf_Piltz\"\u003eAdolf Piltz\u003c/a\u003e."
+ ],
+ "formula": [
+ "If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).",
+ "Columns are multiplicative.",
+ "Dirichlet g.f. for column k: Zeta(s)^k. - _Geoffrey Critzer_, Feb 16 2015"
+ ],
+ "example": [
+ "Rows start:",
+ " 1, 1, 1, 1, 1, 1, 1, ...",
+ " 1, 2, 3, 4, 5, 6, 7, ...",
+ " 1, 2, 3, 4, 5, 6, 7, ...",
+ " 1, 3, 6, 10, 15, 21, 28, ...",
+ " 1, 2, 3, 4, 5, 6, 7, ...",
+ " 1, 4, 9, 16, 25, 36, 49, ...",
+ " ...",
+ "T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - _Geoffrey Critzer_, Feb 16 2015"
+ ],
+ "maple": [
+ "with(numtheory):",
+ "A:= proc(n,k) option remember; `if`(k=1, 1,",
+ " add(A(d, k-1), d=divisors(n)))",
+ " end:",
+ "seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # _Alois P. Heinz_, Feb 25 2015"
+ ],
+ "mathematica": [
+ "tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] \u0026 /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* _Robert G. Wilson v_ *)",
+ "tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] \u0026 /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Sep 13 2020 *)"
+ ],
+ "xref": [
+ "Columns include: A000012, A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221, A111306.",
+ "Rows include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.",
+ "Main diagonal gives A163767.",
+ "Cf. A077593."
+ ],
+ "keyword": "mult,nonn,tabl,look",
+ "offset": "1,5",
+ "author": "_Henry Bottomley_, Nov 08 2002",
+ "ext": [
+ "Typo in formula fixed by _Geoffrey Critzer_, Feb 16 2015"
+ ],
+ "references": 18,
+ "revision": 33,
+ "time": "2020-09-13T02:58:34-04:00",
+ "created": "2003-05-16T03:00:00-04:00"
+ },
+ {
+ "number": 52509,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,4,2,1,1,6,11,8,4,2,1,1,7,16,15,8,4,2,1,1,8,22,26,16,8,4,2,1,1,9,29,42,31,16,8,4,2,1,1,10,37,64,57,32,16,8,4,2,1,1,11,46,93,99,63,32,16,8,4,2,1",
+ "name": "Knights-move Pascal triangle: T(n,k), n \u003e= 0, 0 \u003c= k \u003c= n; T(n,0) = T(n,n) = 1, T(n,k) = T(n-1,k) + T(n-2,k-1) for k = 1,2,...,n-1, n \u003e= 2.",
+ "comment": [
+ "Also square array T(n,k) (n \u003e= 0, k \u003e= 0) read by antidiagonals: T(n,k) = Sum_{i=0..k} binomial(n,i).",
+ "As a number triangle read by rows, this is T(n,k) = Sum_{i=n-2*k..n-k} binomial(n-k,i), with T(n,k) = T(n-1,k) + T(n-2,k-1). Row sums are A000071(n+2). Diagonal sums are A023435(n+1). It is the reverse of the Whitney triangle A004070. - _Paul Barry_, Sep 04 2005",
+ "Also, twice number of orthants intersected by a generic k-dimensional subspace of R^n [Naiman and Scheinerman, 2017]. - _N. J. A. Sloane_, Mar 03 2018"
+ ],
+ "link": [
+ "Reinhard Zumkeller, \u003ca href=\"/A052509/b052509.txt\"\u003eRows n = 0..150 of triangle, flattened\u003c/a\u003e",
+ "C. Kimberling, \u003ca href=\"https://www.fq.math.ca/Scanned/40-4/kimberling.pdf\"\u003ePath-counting and Fibonacci numbers\u003c/a\u003e, Fib. Quart. 40 (4) (2002) 328-338, Example 1C.",
+ "Daniel Q. Naiman, Edward R. Scheinerman, \u003ca href=\"https://arxiv.org/abs/1709.07446\"\u003eArbitrage and Geometry\u003c/a\u003e, arXiv:1709.07446 [q-fin.MF], 2017 [Contains the square array multiplied by 2].",
+ "Richard L. Ollerton and Anthony G. Shannon, \u003ca href=\"http://www.fq.math.ca/Scanned/36-2/ollerton.pdf\"\u003eSome properties of generalized Pascal squares and triangles\u003c/a\u003e, Fib. Q., 36 (1998), 98-109. See Tables 5 and 14.",
+ "D. J. Price, \u003ca href=\"http://www.jstor.org/stable/3609091\"\u003eSome unusual series occurring in n-dimensional geometry\u003c/a\u003e, Math. Gaz., 30 (1946), 149-150.",
+ "\u003ca href=\"/index/Pas#Pascal\"\u003eIndex entries for triangles and arrays related to Pascal's triangle\u003c/a\u003e"
+ ],
+ "formula": [
+ "T(n, k) = Sum_{m=0..n} binomial(n-k, k-m). - _Wouter Meeussen_, Oct 03 2002",
+ "From _Werner Schulte_, Feb 15 2018: (Start)",
+ "Referring to the square array T(i,j):",
+ "G.f. of row n: Sum_{k\u003e=0} T(n,k) * x^k = (1+x)^n / (1-x).",
+ "G.f. of T(i,j): Sum_{k\u003e=0, n\u003e=0} T(n,k) * x^k * y^n = 1 / ((1-x)*(1-y-x*y).",
+ "Let a_i(n) be multiplicative with a_i(p^e) = T(i, e), p prime and e \u003e= 0, then Sum_{n\u003e0} a_i(n)/n^s = (zeta(s))^(i+1) / (zeta(2*s))^i for i \u003e= 0.",
+ "(End)"
+ ],
+ "example": [
+ "Triangle begins:",
+ " 1",
+ " 1, 1",
+ " 1, 2, 1",
+ " 1, 3, 2, 1",
+ " 1, 4, 4, 2, 1",
+ " 1, 5, 7, 4, 2, 1",
+ " 1, 6, 11, 8, 4, 2, 1",
+ "As a square array, this begins:",
+ " 1 1 1 1 1 1 ...",
+ " 1 2 2 2 2 2 ...",
+ " 1 3 4 4 4 4 ...",
+ " 1 4 7 8 8 8 ...",
+ " 1 5 11 15 16 ...",
+ " 1 6 16 26 31 32 ..."
+ ],
+ "maple": [
+ "a := proc(n::nonnegint, k::nonnegint) option remember: if k=0 then RETURN(1) fi: if k=n then RETURN(1) fi: a(n-1,k)+a(n-2,k-1) end: for n from 0 to 11 do for k from 0 to n do printf(`%d,`,a(n,k)) od: od: # _James A. Sellers_, Mar 17 2000",
+ "with(combinat): for s from 0 to 11 do for n from s to 0 by -1 do if n=0 or s-n=0 then printf(`%d,`,1) else printf(`%d,`,sum(binomial(n, i), i=0..s-n)) fi; od: od: # _James A. Sellers_, Mar 17 2000"
+ ],
+ "mathematica": [
+ "Table[Sum[Binomial[n-k, k-m], {m, 0, n}], {n, 0, 10}, {k, 0, n}]"
+ ],
+ "program": [
+ "(PARI) T(n,k)=sum(m=0,n,binomial(n-k,k-m));",
+ "for(n=0,10,for(k=0,n,print1(T(n,k),\", \"););print();); /* show triangle */",
+ "(Haskell)",
+ "a052509 n k = a052509_tabl !! n !! k",
+ "a052509_row n = a052509_tabl !! n",
+ "a052509_tabl = [1] : [1,1] : f [1] [1,1] where",
+ " f row' row = rs : f row rs where",
+ " rs = zipWith (+) ([0] ++ row' ++ [1]) (row ++ [0])",
+ "-- _Reinhard Zumkeller_, Nov 22 2012",
+ "(GAP) A052509:=Flat(List([0..100],n-\u003eList([0..n],k-\u003eSum([0..n],m-\u003eBinomial(n-k,k-m))))); # _Muniru A Asiru_, Sat Feb 17 2018",
+ "(MAGMA) [[(\u0026+[Binomial(n-k, k-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 13 2019",
+ "(Sage) [[sum(binomial(n-k, k-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, May 13 2019"
+ ],
+ "xref": [
+ "Cf. A054123, A054124, A007318, A008949.",
+ "Row sums A000071; Diagonal sums A023435; Mirror A004070.",
+ "Columns give A000027, A000124, A000125, A000127, A006261, ...",
+ "Cf. A052509, A054123, A054124, A007318, A008949, A052553.",
+ "Partial sums across rows of (extended) Pascal's triangle A052553."
+ ],
+ "keyword": "nonn,tabl,easy,nice",
+ "offset": "0,5",
+ "author": "_N. J. A. Sloane_, Mar 17 2000",
+ "ext": [
+ "More terms from _James A. Sellers_, Mar 17 2000",
+ "Entry formed by merging two earlier entries. - _N. J. A. Sloane_, Jun 17 2007",
+ "Edited by _Johannes W. Meijer_, Jul 24 2011"
+ ],
+ "references": 17,
+ "revision": 65,
+ "time": "2019-05-14T02:04:42-04:00",
+ "created": "2000-05-08T03:00:00-04:00"
+ },
+ {
+ "number": 172119,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,4,2,1,1,6,12,8,4,2,1,1,7,20,15,8,4,2,1,1,8,33,28,16,8,4,2,1,1,9,54,52,31,16,8,4,2,1,1,10,88,96,60,32,16,8,4,2,1,1,11,143,177,116,63,32,16,8,4,2,1,1,12,232,326,224,124,64,32,16",
+ "name": "Sum the k preceding elements in the same column and add 1 every time.",
+ "comment": [
+ "Columns are related to Fibonacci n-step numbers. Are there closed forms for the sequences in the columns?",
+ "We denote by a(n,k) the number which is in the (n+1)-th row and (k+1)-th-column. With help of the definition, we also have the recurrence relation: a(n+k+1, k) = 2*a(n+k, k) - a(n, k). We see on the main diagonal the numbers 1,2,4, 8, ..., which is clear from the formula for the general term d(n)=2^n. - _Richard Choulet_, Jan 31 2010",
+ "Most of the paper by Dunkel (1925) is a study of the columns of this table. - _Petros Hadjicostas_, Jun 14 2019"
+ ],
+ "link": [
+ "O. Dunkel, \u003ca href=\"http://www.jstor.org/stable/2298801\"\u003eSolutions of a probability difference equation\u003c/a\u003e, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356.",
+ "T. Langley, J. Liese, and J. Remmel, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html\"\u003eGenerating Functions for Wilf Equivalence Under Generalized Factor Order\u003c/a\u003e, J. Int. Seq. 14 (2011), # 11.4.2.",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/Fibonaccin-StepNumber.html\"\u003eFibonacci n-Step Number\u003c/a\u003e.",
+ "Wikipedia, \u003ca href=\"http://en.wikipedia.org/wiki/Fibonacci_number\"\u003eFibonacci number\u003c/a\u003e."
+ ],
+ "formula": [
+ "T(n,0) = 1.",
+ "T(n,1) = n.",
+ "T(n,2) = A000071(n+1).",
+ "T(n,3) = A008937(n-2).",
+ "The general term in the n-th row and k-th column is given by: a(n, k) = Sum_{j=0..floor(n/(k+1))} ((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)). For example: a(5,3) = binomial(5,5)*2^5 - binomial(2,1)*2^1 = 28. The generating function of the (k+1)-th column satisfies: psi(k)(z)=1/(1-2*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1-z)). - _Richard Choulet_, Jan 31 2010 [By saying \"(k+1)-th column\" the author actually means \"k-th column\" for k = 0, 1, 2, ... - _Petros Hadjicostas_, Jul 26 2019]"
+ ],
+ "example": [
+ "Triangle begins:",
+ "n\\k|....0....1....2....3....4....5....6....7....8....9...10",
+ "---|-------------------------------------------------------",
+ "0..|....1",
+ "1..|....1....1",
+ "2..|....1....2....1",
+ "3..|....1....3....2....1",
+ "4..|....1....4....4....2....1",
+ "5..|....1....5....7....4....2....1",
+ "6..|....1....6...12....8....4....2....1",
+ "7..|....1....7...20...15....8....4....2....1",
+ "8..|....1....8...33...28...16....8....4....2....1",
+ "9..|....1....9...54...52...31...16....8....4....2....1",
+ "10.|....1...10...88...96...60...32...16....8....4....2....1"
+ ],
+ "maple": [
+ "for k from 0 to 20 do for n from 0 to 20 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od: seq(b(n),n=0..20):od; # _Richard Choulet_, Jan 31 2010",
+ "A172119 := proc(n,k)",
+ " option remember;",
+ " if k = 0 then",
+ " 1;",
+ " elif k \u003e n then",
+ " 0;",
+ " else",
+ " 1+add(procname(n-k+i,k),i=0..k-1) ;",
+ " end if;",
+ "end proc:",
+ "seq(seq(A172119(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Sep 16 2017"
+ ],
+ "mathematica": [
+ "T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; k\u003en = 0; T[n_, k_] := T[n, k] = Sum[T[n-k+i, k], {i, 0, k-1}] + 1;",
+ "Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten",
+ "Table[Sum[(-1)^j*2^(n-k-(k+1)*j)*Binomial[n-k-k*j, n-k-(k+1)*j], {j, 0, Floor[(n-k)/(k+1)]}], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 27 2019 *)"
+ ],
+ "program": [
+ "(PARI) T(n,k) = if(k\u003c0 || k\u003en, 0, k==1 \u0026\u0026 k==n, 1, 1 + sum(j=1,k, T(n-j,k)));",
+ "for(n=1,12, for(k=0,n, print1(T(n,k), \", \"))) \\\\ _G. C. Greubel_, Jul 27 2019",
+ "(MAGMA)",
+ "T:= func\u003c n,k | (\u0026+[(-1)^j*2^(n-k-(k+1)*j)*Binomial(n-k-k*j, n-k-(k+1)*j): j in [0..Floor((n-k)/(k+1))]]) \u003e;",
+ "[[T(n,k): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Jul 27 2019",
+ "(Sage)",
+ "@CachedFunction",
+ "def T(n, k):",
+ " if (k==0 and k==n): return 1",
+ " elif (k\u003c0 or k\u003en): return 0",
+ " else: return 1 + sum(T(n-j, k) for j in (1..k))",
+ "[[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 27 2019",
+ "(GAP)",
+ "T:= function(n,k)",
+ " if k=0 and k=n then return 1;",
+ " elif k\u003c0 or k\u003en then return 0;",
+ " else return 1 + Sum([1..k], j-\u003e T(n-j,k));",
+ " fi;",
+ " end;",
+ "Flat(List([0..12], n-\u003e List([0..n], k-\u003e T(n,k) ))); # _G. C. Greubel_, Jul 27 2019"
+ ],
+ "xref": [
+ "Cf. A000071, A008937, A144428.",
+ "Cf. (1-((-1)^T(n, k)))/2 = A051731, see formula by _Hieronymus Fischer_ in A022003."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "0,5",
+ "author": "_Mats Granvik_, Jan 26 2010",
+ "references": 15,
+ "revision": 38,
+ "time": "2019-07-28T17:05:15-04:00",
+ "created": "2010-06-01T03:00:00-04:00"
+ },
+ {
+ "number": 112739,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,5,2,1,1,5,10,7,2,1,1,6,17,22,9,2,1,1,7,26,53,46,11,2,1,1,8,37,106,161,94,13,2,1,1,9,50,187,426,485,190,15,2,1,1,10,65,302,937,1706,1457,382,17,2,1,1,11,82,457,1814,4687,6826,4373,766,19",
+ "name": "Array counting nodes in rooted trees of height n in which the root and internal nodes have valency k (and the leaf nodes have valency one).",
+ "comment": [
+ "Rows of the square array have g.f. (1+x)/((1-x)(1-kx)). They are the partial sums of the coordination sequences for the infinite tree of valency k. Row sums are A112740.",
+ "Rows of the square array are successively: A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023. - _Philippe Deléham_, Feb 22 2014"
+ ],
+ "reference": [
+ "L. He, X. Liu and G. Strang, (2003) Trees with Cantor Eigenvalue Distribution. Studies in Applied Mathematics 110 (2), 123-138.",
+ "L. He, X. Liu and G. Strang, Laplacian eigenvalues of growing trees, Proc. Conf. on Math. Theory of Networks and Systems, Perpignan (2000)."
+ ],
+ "formula": [
+ "As a square array read by antidiagonals, T(n, k)=sum{j=0..k, (2-0^j)*(n-1)^(k-j)}; T(n, k)=(n(n-1)^k-2)/(n-2), n\u003c\u003e2, T(2, n)=2n+1; T(n, k)=sum{j=0..k, (n(n-1)^j-0^j)/(n-1)}, j\u003c\u003e1. As a triangle read by rows, T(n, k)=if(k\u003c=n, sum{j=0..k, (2-0^j)*(n-k-1)^(k-j)}, 0)."
+ ],
+ "example": [
+ "As a square array, rows begin",
+ "1,1,1,1,1,1,... (A000012)",
+ "1,2,2,2,2,2,... (A040000)",
+ "1,3,5,7,9,11,... (A005408)",
+ "1,4,10,22,46,94,... (A033484)",
+ "1,5,17,53,161,485,... (A048473)",
+ "1,6,26,106,426,1706,... (A020989)",
+ "1,7,37,187,937,4687,... (A057651)",
+ "1,8,50,302,1814,10886,... (A061801)",
+ "As a number triangle, rows start",
+ "1;",
+ "1,1;",
+ "1,2,1;",
+ "1,3,2,1;",
+ "1,4,5,2,1;",
+ "1,5,10,7,2,1;"
+ ],
+ "xref": [
+ "Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023."
+ ],
+ "keyword": "easy,nonn,tabl",
+ "offset": "0,5",
+ "author": "_Paul Barry_, Sep 16 2005",
+ "references": 10,
+ "revision": 10,
+ "time": "2014-02-23T12:39:14-05:00",
+ "created": "2005-09-21T03:00:00-04:00"
+ },
+ {
+ "number": 194005,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,3,3,1,1,5,4,6,3,1,1,6,5,10,6,4,1,1,7,6,15,10,10,4,1,1,8,7,21,15,20,10,5,1,1,9,8,28,21,35,20,15,5,1,1,10,9,36,28,56,35,35,15,6,1,1,11,10,45,36,84,56,70,35,21,6,1",
+ "name": "Triangle of the coefficients of an (n+1)-th order differential equation associated with A103631.",
+ "comment": [
+ "This triangle is a companion to Parks' triangle A103631.",
+ "The coefficients of triangle A103631(n,k) appear in appendix 2 of Park’s remarkable article “A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov” if we assume that the b(n) coefficients are all equal to 1, see the second Maple program.",
+ "The a(n,k) coefficients of the triangle given above are related to the coefficients of a linear (n+1)-th order differential equation for the case b(n)=1, see the examples.",
+ "a(n,k) is also the number of symmetric binary strings of odd length n with Hamming weight k\u003e0 and no consecutive 1's. - _Christian Barrientos_ and _Sarah Minion_, Feb 27 2018"
+ ],
+ "link": [
+ "Reinhard Zumkeller, \u003ca href=\"/A194005/b194005.txt\"\u003eRows n = 0..150 of triangle, flattened\u003c/a\u003e",
+ "Henry W. Gould, \u003ca href=\"http://www.fq.math.ca/3-4.html\"\u003e A Variant of Pascal's Triangle \u003c/a\u003e, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271.",
+ "P.C. Parks, \u003ca href=\"https://doi.org/10.1017/S030500410004072X\"\u003e A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov \u003c/a\u003e, Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702.",
+ "Chris Zheng, Jeffrey Zheng, \u003ca href=\"https://doi.org/10.1007/978-981-13-2282-2_4\"\u003eTriangular Numbers and Their Inherent Properties\u003c/a\u003e, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.",
+ "\u003ca href=\"/index/Pas#Pascal\"\u003eIndex entries for triangles and arrays related to Pascal's triangle\u003c/a\u003e"
+ ],
+ "formula": [
+ "a(n,k) = binomial(floor((2*n+1-k)/2), n-k).",
+ "a(n,k) = sum(A103631(n1,k), n1=k..n), 0\u003c=k\u003c=n and n\u003e=0.",
+ "a(n,k) = sum(binomial(floor((2*n1-k-1)/2), n1-k), n1=k..n).",
+ "T(n,0) = T(n,n) = 1, T(n,k) = T(n-2,k-2) + T(n-1,k), 0 \u003c k \u003c n. - _Reinhard Zumkeller_, Nov 23 2012"
+ ],
+ "example": [
+ "For the 5th-order linear differential equation the coefficients a(k) are: a(0) = 1, a(1) = a(4,0) = 1, a(2) = a(4,1) = 4, a(3) = a(4,2) = 3, a(4) = a(4,3) = 3 and a(5) = a(4,4) = 1.",
+ "The corresponding Hurwitz matrices A(k) are, see Parks: A(5) = Matrix([[a(1),a(0),0,0,0], [a(3),a(2),a(1),a(0),0], [a(5),a(4),a(3),a(2),a(1)], [0,0,a(5),a(4),a(3)], [0,0,0,0,a(5)]]), A(4) = Matrix([[a(1),a(0),0,0], [a(3),a(2),a(1),a(0)], [a(5),a(4),a(3),a(2)], [0,0,a(5),a(4)]]), A(3) = Matrix([[a(1),a(0),0], [a(3),a(2),a(1)], [a(5),a(4),a(3)]]), A(2) = Matrix([[a(1),a(0)], [a(3),a(2)]]) and A(1) = Matrix([[a(1)]]).",
+ "The values of b(k) are, see Parks: b(1) = d(1), b(2) = d(2)/d(1), b(3) = d(3)/(d(1)*d(2)), b(4) = d(1)*d(4)/(d(2)*d(3)) and b(5) = d(2)*d(5)/(d(3)*d(4)).",
+ "These a(k) values lead to d(k) = 1 and subsequently to b(k) = 1 and this confirms our initial assumption, see the comments."
+ ],
+ "maple": [
+ "A194005 := proc(n,k): binomial(floor((2*n+1-k)/2),n-k) end: for n from 0 to 11 do seq(A194005(n,k), k=0..n) od; seq(seq(A194005(n,k), k=0..n), n=0..11);",
+ "nmax:=11: for n from 0 to nmax+1 do b(n):=1 od: A103631 := proc(n,k) option remember: local j: if k=0 and n=0 then b(1) elif k=0 and n\u003e=1 then 0 elif k=1 then b(n+1) elif k=2 then b(1)*b(n+1) elif k\u003e=3 then expand(b(n+1)*add(procname(j,k-2), j=k-2..n-2)) fi: end: for n from 0 to nmax do for k from 0 to n do A194005(n,k):= add(A103631(n1,k), n1=k..n) od: od: seq(seq(A194005(n,k),k=0..n), n=0..nmax);"
+ ],
+ "mathematica": [
+ "Flatten[Table[Binomial[Floor[(2n+1-k)/2],n-k],{n,0,20},{k,0,n}]] (* _Harvey P. Dale_, Apr 15 2012 *)"
+ ],
+ "program": [
+ "(Haskell)",
+ "a194005 n k = a194005_tabl !! n !! k",
+ "a194005_row n = a194005_tabl !! n",
+ "a194005_tabl = [1] : [1,1] : f [1] [1,1] where",
+ " f row' row = rs : f row rs where",
+ " rs = zipWith (+) ([0,1] ++ row') (row ++ [0])",
+ "-- _Reinhard Zumkeller_, Nov 22 2012"
+ ],
+ "xref": [
+ "Cf. A065941 and A103631.",
+ "Triangle sums (see A180662): A000071 (row sums; alt row sums), A075427 (Kn22), A000079 (Kn3), A109222(n+1)-1 (Kn4), A000045 (Fi1), A034943 (Ca3), A001519 (Gi3), A000930 (Ze3)",
+ "Interesting diagonals: T(n,n-4) = A189976(n+5) and T(n,n-5) = A189980(n+6)",
+ "Cf. A052509."
+ ],
+ "keyword": "nonn,easy,tabl",
+ "offset": "0,5",
+ "author": "_Johannes W. Meijer_ \u0026 A. Hirschberg (a.hirschberg(AT)tue.nl), Aug 11 2011",
+ "references": 9,
+ "revision": 48,
+ "time": "2019-02-07T19:31:52-05:00",
+ "created": "2011-08-11T17:58:56-04:00"
+ },
+ {
+ "number": 308813,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,5,3,1,1,5,10,11,2,1,1,6,17,31,17,4,1,1,7,26,69,82,39,2,1,1,8,37,131,257,256,65,4,1,1,9,50,223,626,1045,730,139,3,1,1,10,65,351,1297,3156,4097,2218,261,4,1",
+ "name": "Square array A(n,k), n \u003e= 1, k \u003e= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} k^(d-1).",
+ "link": [
+ "Seiichi Manyama, \u003ca href=\"/A308813/b308813.txt\"\u003eAntidiagonals n = 1..140, flattened\u003c/a\u003e"
+ ],
+ "formula": [
+ "G.f. of column k: Sum_{j\u003e=1} x^j/(1 - k*x^j)."
+ ],
+ "example": [
+ "Square array begins:",
+ " 1, 1, 1, 1, 1, 1, 1, ...",
+ " 1, 2, 3, 4, 5, 6, 7, ...",
+ " 1, 2, 5, 10, 17, 26, 37, ...",
+ " 1, 3, 11, 31, 69, 131, 223, ...",
+ " 1, 2, 17, 82, 257, 626, 1297, ...",
+ " 1, 4, 39, 256, 1045, 3156, 7819, ...",
+ " 1, 2, 65, 730, 4097, 15626, 46657, ..."
+ ],
+ "xref": [
+ "Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.",
+ "Row n=1..3 give A000012, A000027(n+1), A002522.",
+ "A(n,n) gives A308814."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "1,5",
+ "author": "_Seiichi Manyama_, Jun 26 2019",
+ "references": 7,
+ "revision": 31,
+ "time": "2020-12-14T09:36:44-05:00",
+ "created": "2019-06-26T18:07:35-04:00"
+ },
+ {
+ "number": 55794,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,0,1,5,7,4,1,0,1,6,11,8,3,0,0,1,7,16,15,7,1,0,0,1,8,22,26,15,4,0,0,0,1,9,29,42,30,11,1,0,0,0,1,10,37,64,56,26,5,0,0,0,0,1,11,46,93,98,56,16,1,0,0,0,0,1,12,56,130,162,112,42,6,0,0,0,0,0",
+ "name": "Triangle T read by rows: T(i,0)=1 for i \u003e= 0; T(i,i)=0 for i=0,1,2,3; T(i,i)=0 for i \u003e= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1\u003c=j\u003c=i-1.",
+ "comment": [
+ "T(i+j,j) is the number of strings (s(1),...,s(i+1)) of nonnegative integers s(k) such that 0\u003c=s(k)-s(k-1)\u003c=1 for k=2,3,...,i+1 and s(i+1)=j.",
+ "T(i+j,j) is the number of compositions of j consisting of i parts, all of in {0,1}."
+ ],
+ "link": [
+ "G. C. Greubel, \u003ca href=\"/A055794/b055794.txt\"\u003eRows n = 0..100 of triangle, flattened\u003c/a\u003e",
+ "C. Kimberling, \u003ca href=\"https://www.fq.math.ca/Scanned/40-4/kimberling.pdf\"\u003ePath-counting and Fibonacci numbers\u003c/a\u003e, Fib. Quart. 40 (4) (2002) 328-338, Example 1B."
+ ],
+ "example": [
+ "Triangle begins:",
+ " 1;",
+ " 1, 1;",
+ " 1, 2, 1;",
+ " 1, 3, 2, 1;",
+ " 1, 4, 4, 2, 0;",
+ " 1, 5, 7, 4, 1, 0;",
+ " ...",
+ "T(7,4) counts the strings 3334, 3344, 3444, 2234, 2334, 2344, 1234.",
+ "T(7,4) counts the compositions 001, 010, 100, 011, 101, 110, 111."
+ ],
+ "maple": [
+ "T:= proc(n, k) option remember;",
+ " if k=0 then 1",
+ " elif k=n and n\u003c4 then 1",
+ " elif k=n then 0",
+ " else T(n-1, k) + T(n-2, k-1)",
+ " fi; end:",
+ "seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 25 2020"
+ ],
+ "mathematica": [
+ "T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n \u0026\u0026 n\u003c4, 1, If[k==n, 0, T[n-1, k] + T[n-2, k-1] ]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 25 2020 *)"
+ ],
+ "program": [
+ "(PARI) T(n,k) = if(k==0, 1, if(k==n \u0026\u0026 n\u003c4, 1, if(k==n, 0, T(n-1, k) + T(n-2, k-1) )));",
+ "for(n=0,12, for(k=0,n, print1(T(n,k), \", \"))) \\\\ _G. C. Greubel_, Jan 25 2020",
+ "(MAGMA)",
+ "function T(n,k)",
+ " if k eq 0 then return 1;",
+ " elif k eq n and n lt 4 then return 1;",
+ " elif k eq n then return 0;",
+ " else return T(n-1,k) + T(n-2, k-1);",
+ " end if; return T; end function;",
+ "[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 25 2020",
+ "(Sage)",
+ "@CachedFunction",
+ "def T(n, k):",
+ " if (k==0): return 1",
+ " elif (k==n and n\u003c4): return 1",
+ " elif (k==n): return 0",
+ " else: return T(n-1, k) + T(n-2, k-1)",
+ "[[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 25 2020",
+ "(GAP)",
+ "T:= function(n,k)",
+ " if k=0 then return 1;",
+ " elif k=n and n\u003c4 then return 1;",
+ " elif k=n then return 0;",
+ " else return T(n-1,k) + T(n-2,k-1);",
+ " fi; end;",
+ "Flat(List([0..12], n-\u003e List([0..n], k-\u003e T(n,k) ))); # _G. C. Greubel_, Jan 25 2020"
+ ],
+ "xref": [
+ "Row sums: A000032 (Lucas numbers, 1, 2, 4, 7, 11, 18, ...).",
+ "T(2n, n)=A000125(n) (Cake numbers, 1, 2, 4, 8, 15, 26, ...).",
+ "T(2n+2, n)=A027660(n)."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "0,5",
+ "author": "_Clark Kimberling_, May 28 2000",
+ "references": 6,
+ "revision": 15,
+ "time": "2020-01-25T20:57:23-05:00",
+ "created": "2000-06-15T03:00:00-04:00"
+ },
+ {
+ "number": 92905,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,6,4,2,1,1,6,9,7,4,2,1,1,7,12,11,7,4,2,1,1,8,16,16,12,7,4,2,1,1,9,20,23,18,12,7,4,2,1,1,10,25,31,27,19,12,7,4,2,1,1,11,30,41,38,29,19,12,7,4,2,1,1,12,36,53,53,42,30,19,12,7,4,2,1",
+ "name": "Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n\u003e=0, where each row begins with 1.",
+ "comment": [
+ "Row sums form A000070, which is the partial sums of the partition numbers (A000041). Rows read backwards converge to the row sums (A000070).",
+ "Contribution from _Alford Arnold_, Feb 07 2010: (Start)",
+ "The table can also be generated by summing sequences embedded within Table A008284",
+ "For example,",
+ "1 1 1 1 ... yields 1 2 3 4 ...",
+ "1 1 2 2 3 3 ... yields 1 2 4 6 9 12 ...",
+ "1 1 2 3 4 5 7 ... yields 1 2 4 7 11 16 ...",
+ "(End)",
+ "T(n,k) is also count of all 'replacable' cells in the (Ferrers plots of) the partitions on n in exactly k parts. [From _Wouter Meeussen_, Sep 16 2010]",
+ "From _Wolfdieter Lang_, Dec 03 2012 (Start)",
+ "The triangle entry T(n,k) is obtained from triangle A072233 by summing the entries of column k up to n (see the partial sum type o.g.f. given by Vladeta Jovovic in the formula section).",
+ " Therefore, the o.g.f. for the sequence in column k is x^k/((1-x)* product(1-x^j,j=1..k)).",
+ "The triangle with entry a(n,m) = T(n-1,m-1), n \u003e= 1, m = 1, ..., n, is obtained from the partition array A103921 when in row n all entries belonging to part number m are summed (a conjecture). (End)"
+ ],
+ "link": [
+ "V. V. Kruchinin, \u003ca href=\"https://doi.org/10.1134/S0001434609090260\"\u003eThe number of partitions of a natural number n into parts each of which is not less than m\u003c/a\u003e, Math. Notes 86 (4) (2009) 505-509",
+ "R. J. Mathar, \u003ca href=\"http://www.mpia.de/~mathar/public/mathar20171110.pdf\"\u003eSize of the set of residues of integer powers of fixed exponent\u003c/a\u003e, (2017), Table 11."
+ ],
+ "formula": [
+ "T(n, k) = sum_{j=0..k} T(n-k, j), with T(n, 0) = 1 for all n\u003e=0. A000070(n) = sum_{k=0..n} T(n, k).",
+ "O.g.f.: (1/(1-y))*(1/Product(1-x*y^k, k=1..infinity)). - _Vladeta Jovovic_, Jan 29 2005"
+ ],
+ "example": [
+ "The fourth row (n=3) is {1,3,2,1} and the fourth diagonal is the partial sums of the fourth row: {1,4,6,7,7,7,7,7,...}.",
+ "The triangle T(n,k) begins:",
+ "n\\k 0 1 2 3 4 5 6 7 8 9 10 11 12 ...",
+ "0 1",
+ "1 1 1",
+ "2 1 2 1",
+ "3 1 3 2 1",
+ "4 1 4 4 2 1",
+ "5 1 5 6 4 2 1",
+ "6 1 6 9 7 4 2 1",
+ "7 1 7 12 11 7 4 2 1",
+ "8 1 8 16 16 12 7 4 2 1",
+ "9 1 9 20 23 18 12 7 4 2 1",
+ "10 1 10 25 31 27 19 12 7 4 2 1",
+ "11 1 11 30 41 38 29 19 12 7 4 2 1",
+ "12 1 12 36 53 53 42 30 19 12 7 4 2 1",
+ "... Reformatted by _Wolfdieter Lang_, Dec 03 2012",
+ "T(5,3)=4 because the partitions of 5 in exactly 3 parts are 221 and 311, and they give rise to partitions of 4 in four ways: 221-\u003e22 and 211, 311-\u003e211 and 31, since both their Ferrers plots have 2 'mobile cells' each. [From _Wouter Meeussen_, Sep 16 2010]",
+ "T(5,3) = a(6,4) = 4 because the partitions of 6 with 4 parts are 1113 and 1122, with the number of distinct parts 2 and 2, respectively, summing to 4 (see the array A103921). An example for the conjecture given as comment above. - _Wolfdieter Lang_, Dec 03 2012"
+ ],
+ "maple": [
+ "T(n,k)=if(n\u003ck|k\u003c0,0,if(n==k|k==0,1,sum(j=0,min(k,n-k),T(n-k,j))))"
+ ],
+ "mathematica": [
+ "(*Needs[\"DiscreteMath`Combinatorica`\"]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] \u0026 ) /@ Partitions[n - m, m] *); mobile[p_?PartitionQ]:=1+Count[Drop[p,-1]-Rest[p],_?Positive]; Table[Tr[mobile/@partitionexact[n,k]],{n,12},{k,n}] [From _Wouter Meeussen_, Sep 16 2010]"
+ ],
+ "xref": [
+ "Antidiagonal sums form the partition numbers (A000041).",
+ "Cf. A000070.",
+ "Cf. A008284 [From _Alford Arnold_, Feb 07 2010]",
+ "Columns: A087811, A000601, A002621, A002622, A288341 - A288345."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "0,5",
+ "author": "_Paul D. Hanna_, Mar 12 2004",
+ "ext": [
+ "Several corrections by _Wolfdieter Lang_, Dec 03 2012."
+ ],
+ "references": 6,
+ "revision": 25,
+ "time": "2019-03-28T10:19:35-04:00",
+ "created": "2004-06-12T03:00:00-04:00"
+ },
+ {
+ "number": 228125,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,5,2,1,1,6,10,9,5,2,1,1,7,14,16,10,5,2,1,1,8,19,24,19,11,5,2,1,1,9,24,37,32,21,11,5,2,1,1,10,30,51,52,38,22,11,5,2,1,1,11,37,71,79,66,41,23,11,5,2,1,1,12,44,93,117,106,74,43,23,11,5,2,1,1,13,52,122,166,166,125,80,44,23,11,5,2,1,1,14,61,153,231,251,204,139,83,45,23,11,5,2,1,1,15,70,193,311,367,322,236,147,85,45,23,11,5,2,1",
+ "name": "Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.",
+ "comment": [
+ "Row sums equal A003293.",
+ "Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ..."
+ ],
+ "example": [
+ "T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are",
+ "114 , 123 , 222 , 11 , 12 , 13 , 1",
+ " 4 3 2 2",
+ " 3",
+ "Triangle starts:",
+ "1;",
+ "1, 1;",
+ "1, 2, 1;",
+ "1, 3, 2, 1;",
+ "1, 4, 4, 2, 1;",
+ "1, 5, 7, 5, 2, 1;",
+ "1, 6, 10, 9, 5, 2, 1;",
+ "1, 7, 14, 16, 10, 5, 2, 1;",
+ "1, 8, 19, 24, 19, 11, 5, 2, 1;",
+ "1, 9, 24, 37, 32, 21, 11, 5, 2, 1;",
+ "1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;"
+ ],
+ "mathematica": [
+ "hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q\u003e=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];",
+ "Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1\u0026) /@ Flatten[hooklength[#1]],{q,0,w}]\u0026) /@ Partitions[n]],{w,24},{n,w}]"
+ ],
+ "xref": [
+ "Cf. A003293, A005986."
+ ],
+ "keyword": "nonn,tabl",
+ "offset": "1,5",
+ "author": "_Wouter Meeussen_, Aug 11 2013",
+ "references": 5,
+ "revision": 21,
+ "time": "2016-07-31T09:03:15-04:00",
+ "created": "2013-08-13T03:37:46-04:00"
+ },
+ {
+ "number": 227588,
+ "data": "1,1,1,1,2,1,1,3,2,1,1,4,4,2,1,1,5,7,5,2,1,1,6,12,12,6,2,1,1,7,18,24,16,7,2,1,1,8,26,46,42,23,8,2,1,1,9,35,83,101,73,29,9,2,1",
+ "name": "Maximum label within a minimal labeling of k \u003e= 0 identical n-sided dice (n \u003e= 1) yielding the most possible sums; square array A(n,k), read by antidiagonals.",
+ "link": [
+ "The \u003ca href=\"http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/Challenges/July2013.html\"\u003eIBM Ponder This July 2013\u003c/a\u003e challenge asks for A(8,3)."
+ ],
+ "example": [
+ "Three tetrahedra labeled (1, 2, 8, 12) yield the 20 possible sums 3, 4, 5, 6, 10, 11, 12, 14, 15, 16, 17, 18, 21, 22, 24, 25, 26, 28, 32, 36. No more sums can be obtained by different labelings, and no labeling with labels \u003c 12 yields 20 possible sums. Therefore A(4,3) = 12.",
+ "Square array A(n,k) begins:",
+ "1, 1, 1, 1, 1, 1, 1, 1, ...",
+ "1, 2, 2, 2, 2, 2, 2, ...",
+ "1, 3, 4, 5, 6, 7, ...",
+ "1, 4, 7, 12, 16, ...",
+ "1, 5, 12, 24, ...",
+ "1, 6, 18, ...",
+ "1, 7, ...",
+ "1, ..."
+ ],
+ "xref": [
+ "Cf. A227589, A227590, A227358."
+ ],
+ "keyword": "nonn,tabl,more",
+ "offset": "1,5",
+ "author": "_Jens Voß_, Jul 17 2013",
+ "references": 4,
+ "revision": 18,
+ "time": "2013-09-13T21:00:37-04:00",
+ "created": "2013-07-18T10:56:21-04:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,1,2,1,6,2,.json b/data/1,1,2,1,6,2,.json
new file mode 100644
index 0000000..50e9bf6
--- /dev/null
+++ b/data/1,1,2,1,6,2,.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,2,1,6,2,24,6,1,120,24,3,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/1,1,3,6,15,3.json b/data/1,1,3,6,15,3.json
new file mode 100644
index 0000000..972bade
--- /dev/null
+++ b/data/1,1,3,6,15,3.json
@@ -0,0 +1,287 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,1,3,6,15,36,91,232,603,1585,4213,11298,",
+ "count": 3,
+ "start": 0,
+ "results": [
+ {
+ "number": 5043,
+ "id": "M2587",
+ "data": "1,0,1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,1730787,4805595,13393689,37458330,105089229,295673994,834086421,2358641376,6684761125,18985057351,54022715451,154000562758,439742222071,1257643249140",
+ "name": "Riordan numbers: a(n) = (n-1)*(2*a(n-1) + 3*a(n-2))/(n+1).",
+ "comment": [
+ "Also called Motzkin summands or ring numbers.",
+ "The old name was \"Motzkin sums\", used in certain publications. The sequence has the property that Motzkin(n) = A001006(n) = a(n) + a(n+1), e.g., A001006(4) = 9 = 3 + 6 = a(4) + a(5).",
+ "Number of 'Catalan partitions', that is partitions of a set 1,2,3,...,n into parts that are not singletons and whose convex hulls are disjoint when the points are arranged on a circle (so when the parts are all pairs we get Catalan numbers). - Aart Blokhuis (aartb(AT)win.tue.nl), Jul 04 2000",
+ "Number of ordered trees with n edges and no vertices of outdegree 1. For n \u003e 1, number of dissections of a convex polygon by nonintersecting diagonals with a total number of n+1 edges. - _Emeric Deutsch_, Mar 06 2002",
+ "Number of Motzkin paths of length n with no horizontal steps at level 0. - _Emeric Deutsch_, Nov 09 2003",
+ "Number of Dyck paths of semilength n with no peaks at odd level. Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1), D=(1,-1). Number of Dyck paths of semilength n with no ascents of length 1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUUDDD. - _Emeric Deutsch_, Dec 05 2003",
+ "Arises in Schubert calculus as follows. Let P = complex projective space of dimension n+1. Take n projective subspaces of codimension 3 in P in general position. Then a(n) is the number of lines of P intersecting all these subspaces. - F. Hirzebruch, Feb 09 2004",
+ "Difference between central trinomial coefficient and its predecessor. Example: a(6) = 15 = 141 - 126 and (1 + x + x^2)^6 = ... + 126*x^5 + 141*x^6 + ... (Catalan number A000108(n) is the difference between central binomial coefficient and its predecessor.) - _David Callan_, Feb 07 2004",
+ "a(n) = number of 321-avoiding permutations on [n] in which each left-to-right maximum is a descent (i.e., is followed by a smaller number). For example, a(4) counts 4123, 3142, 2143. - _David Callan_, Jul 20 2005",
+ "The Hankel transform of this sequence give A000012 = [1, 1, 1, 1, 1, 1, 1, ...]; example: Det([1, 0, 1, 1; 0, 1, 1, 3; 1, 1, 3, 6; 1, 3, 6, 15]) = 1. - _Philippe Deléham_, May 28 2005",
+ "The number of projective invariants of degree 2 for n labeled points on the projective line. - Benjamin J. Howard (bhoward(AT)ima.umn.edu), Nov 24 2006",
+ "Define a random variable X=trA^2, where A is a 2 X 2 unitary symplectic matrix chosen from USp(2) with Haar measure. The n-th central moment of X is E[(X+1)^n] = a(n). - _Andrew V. Sutherland_, Dec 02 2007",
+ "Let V be the adjoint representation of the complex Lie algebra sl(2). The dimension of the invariant subspace of the n-th tensor power of V is a(n). - Samson Black (sblack1(AT)uoregon.edu), Aug 27 2008",
+ "Starting with offset 3 = iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - _Gary W. Adamson_, Jan 08 2009",
+ "a(n) has the following standard-Young-tableaux (SYT) interpretation: binomial(n+1,k)*binomial(n-k-1,k-1)/(n+1)=f^(k,k,1^{n-2k}) where f^lambda equals the number of SYT of shape lambda. - Amitai Regev (amotai.regev(AT)weizmann.ac.il), Mar 02 2010",
+ "a(n) is also the sum of the numbers of standard Young tableaux of shapes (k,k,1^{n-2k}) for all 1 \u003c= k \u003c= floor(n/2). - Amitai Regev (amotai.regev(AT)weizmann.ac.il), Mar 10 2010",
+ "a(n) is the number of derangements of {1,2,...,n} having genus 0. The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q. Example: a(3)=1 because p=231=(123) is the only derangement of {1,2,3} with genus 0. Indeed, cp'=231*312=123=(1)(2)(3) and so g(p)= (1/2)(3+1-1-3)=0. - _Emeric Deutsch_, May 29 2010",
+ "Apparently: Number of Dyck 2n-paths with all ascents length 2 and no descent length 2. - _David Scambler_, Apr 17 2012",
+ "This is true. Proof: The mapping \"insert a peak (UD) after each upstep (U)\" is a bijection from all Dyck n-paths to those Dyck (2n)-paths in which each ascent is of length 2. It sends descents of length 1 in the n-path to descents of length 2 in the (2n)-path. But Dyck n-paths with no descents of length 1 are equinumerous with Riordan n-paths (Motzkin n-paths with no flatsteps at ground level) as follows. Given a Dyck n-path with no descents of length 1, split it into consecutive step pairs, then replace UU with U, DD with D, UD with a blue flatstep (F), DU with a red flatstep, and concatenate the new steps to get a colored Motzkin path. Each red F will be (immediately) preceded by a blue F or a D. In the latter case, transfer the red F so that it precedes the matching U of the D. Finally, erase colors to get the required Riordan path. For example, with lowercase f denoting a red flatstep, U^5 D^2 U D^4 U^4 D^3 U D^2 -\u003e (U^2, U^2, UD, DU, D^2, D^2, U^2, U^2 D^2, DU, D^2) -\u003e UUFfDDUUDfD -\u003e UUFFDDUFUDD. - _David Callan_, Apr 25 2012",
+ "From _Nolan Wallach_, Aug 20 2014: (Start)",
+ "Let ch[part1, part2] be the value of the character of the symmetric group on n letters corresponding to the partition part1 of n on the conjucgacy class given by part2. Let A[n] be the set of (n+1) partitions of 2n with parts 1 or 2. Then deleting the first term of the sequence one has a(n) = Sum_{k=1..n+1} binomial(n,k-1)*ch[[n,n], A[n][[k]]])/2^n. This via the Frobenius Character Formula can be interpreted as the dimension of the SL(n,C) invariants in tensor^n (wedge^2 C^n).",
+ "Explanation: Let p_j denote sum (x_i)^j the sum in k variables. Then the Frobenius formula says then (p_1)^j_1 (p_2)^j_2 ... (p_r)^j_r is equal to sum(lambda, ch[lambda, 1^j_12^j_2 ... r^j_r] S_lambda) with S_lambda the Shur function corresponding to lambda. This formula implies that the coefficient of S([n,n]) in (((p_1)^1+p_2)/2)^n in its expansion in terms of Shur functions is the right hand side of our formula. If we specialize the number of variables to 2 then S[n,n](x,y)=(xy)^n. Which when restricted to y=x^(-1) is 1. That is it is 1 on SL(2).",
+ "On the other hand ((p_1)^2+p_2)/2 is the complete homogeneous symmetric function of degree 2 that is tr(S^2(X)). Thus our formula for a(n) is the same as that of Samson Black above since his V is the same as S^2(C^2) as a representation of SL(2). On the other hand, if we multiply ch(lambda) by sgn you get ch(Transpose(lambda)). So ch([n,n]) becomes ch([2,...,2]) (here there are n 2's). The formula for a(n) is now (1/2^n)*Sum_{j=0..n} ch([2,..,2], 1^(2n-2j) 2^j])*(-1)^j)*binomial(n,j), which calculates the coefficient of S_(2,...,2) in (((p_1)^2-p_2)/2)^n. But ((p_1)^2-p_2)/2 in n variables is the second elementary symmetric function which is the character of wedge^2 C^n and S_(2,...,2) is 1 on SL(n).",
+ "(End)",
+ "a(n) = number of noncrossing partitions (A000108) of [n] that contain no singletons, also number of nonnesting partitions (A000108) of [n] that contain no singletons. - _David Callan_, Aug 27 2014",
+ "From _Tom Copeland_, Nov 02 2014: (Start)",
+ "Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).",
+ "Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].",
+ "Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).",
+ "BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)].",
+ "Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).",
+ "Various relations among the o.g.f.s may be easily constructed, such as Fib[-Mot(-x)] = -P[P(-x)] = x/(1-2*x) a generating fct for 2^n.",
+ "Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1]. (End)",
+ "Consistent with David Callan's comment above, A249548, provides a refinement of the Motzkin sums into the individual numbers for the non-crossing partitions he describes. - _Tom Copeland_, Nov 09 2014",
+ "The number of lattice paths from (0,0) to (n,0) that do not cross below the x-axis and use up-step=(1,1) and down-steps=(1,-k) where k is a positive integer. For example, a(4) = 3: [(1,1)(1,1)(1,-1)(1,-1)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,1)(1,-3)]. - _Nicholas Ham_, Aug 19 2015",
+ "A series created using 2*(a(n) + a(n+1)) + (a(n+1) + a(n+2)) has Hankel transform of F(2n), offset 3, F being a Fibonacci number, A001906 (Empirical observation). - _Tony Foster III_, Jul 30 2016",
+ "The series a(n) + A001006(n) has Hankel transform F(2n+1), offset n=1, F being the Fibonacci bisection A001519 (empirical observation). - _Tony Foster III_, Sep 05 2016",
+ "The Rubey and Stump reference proves a refinement of a conjecture of René Marczinzik, which they state as: \"The number of 2-Gorenstein algebras which are Nakayama algebras with n simple modules and have an oriented line as associated quiver equals the number of Motzkin paths of length n. Moreover, the number of such algebras having the double centraliser property with respect to a minimal faithful projective-injective module equals the number of Riordan paths, that is, Motzkin paths without level-steps at height zero, of length n.\" - _Eric M. Schmidt_, Dec 16 2017",
+ "A connection to the Thue-Morse sequence: (-1)^a(n) = (-1)^A010060(n) * (-1)^A010060(n+1) = A106400(n) * A106400(n+1). - _Vladimir Reshetnikov_, Jul 21 2019",
+ "Named by Bernhart (1999) after the American mathematician John Riordan (1903-1988). - _Amiram Eldar_, Apr 15 2021"
+ ],
+ "reference": [
+ "N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).",
+ "D. N. Verma, Towards Classifying Finite Point-Set Configurations, preprint, 1997. [Apparently unpublished]"
+ ],
+ "link": [
+ "G. C. Greubel, \u003ca href=\"/A005043/b005043.txt\"\u003eTable of n, a(n) for n = 0..1000\u003c/a\u003e (terms 0..200 from T. D. Noe)",
+ "Prarit Agarwal and June Nahmgoong, \u003ca href=\"https://arxiv.org/abs/2001.10826\"\u003eSinglets in the tensor product of an arbitrary number of Adjoint representations of SU(3)\u003c/a\u003e, arXiv:2001.10826 [math.RT], 2020.",
+ "O. Aichholzer, A. Asinowski, and T. Miltzow, \u003ca href=\"http://arxiv.org/abs/1403.5546\"\u003eDisjoint compatibility graph of non-crossing matchings of points in convex position\u003c/a\u003e, arXiv preprint arXiv:1403.5546 [math.CO], 2014.",
+ "M. Aigner, \u003ca href=\"http://dx.doi.org/10.1016/j.disc.2007.06.012\"\u003eEnumeration via ballot numbers\u003c/a\u003e, Discrete Math., 308 (2008), 2544-2563.",
+ "Gert Almkvist, Warren Dicks, and Edward Formanek, \u003ca href=\"http://dx.doi.org/10.1016/0021-8693(85)90183-8\"\u003eHilbert series of fixed free algebras and noncommutative classical invariant theory\u003c/a\u003e, J. Algebra 93 (1985), no. 1, 189-214.",
+ "D. L. Andrews, \u003ca href=\"/A005043/a005043.pdf\"\u003eLetter to N. J. A. Sloane\u003c/a\u003e, Apr 10 1978.",
+ "D. L. Andrews and T. Thirunamachandran, \u003ca href=\"http://dx.doi.org/10.1063/1.434725\"\u003eOn three-dimensional rotational averages\u003c/a\u003e, J. Chem. Phys., 67 (1977), 5026-5033.",
+ "D. L. Andrews and T. Thirunamachandran, \u003ca href=\"/A005043/a005043_1.pdf\"\u003eOn three-dimensional rotational averages\u003c/a\u003e, J. Chem. Phys., 67 (1977), 5026-5033. [Annotated scanned copy]",
+ "J.-L. Baril, \u003ca href=\"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178\"\u003eClassical sequences revisited with permutations avoiding dotted pattern\u003c/a\u003e, Electronic Journal of Combinatorics, 18 (2011), #P178.",
+ "Paul Barry, \u003ca href=\"https://arxiv.org/abs/1802.03443\"\u003eOn a transformation of Riordan moment sequences\u003c/a\u003e, arXiv:1802.03443 [math.CO], 2018.",
+ "Paul Barry, \u003ca href=\"https://arxiv.org/abs/1912.01126\"\u003eRiordan arrays, the A-matrix, and Somos 4 sequences\u003c/a\u003e, arXiv:1912.01126 [math.CO], 2019.",
+ "Paul Barry, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL23/Barry/barry444.html\"\u003eOn the Central Antecedents of Integer (and Other) Sequences\u003c/a\u003e, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.",
+ "Paul Barry and Aoife Hennessy, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry1/barry202.html\"\u003eFour-term Recurrences, Orthogonal Polynomials and Riordan Arrays\u003c/a\u003e, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012",
+ "Frank R. Bernhart, \u003ca href=\"http://dx.doi.org/10.1016/S0012-365X(99)00054-0\"\u003eCatalan, Motzkin and Riordan numbers\u003c/a\u003e, Discr. Math., 204 (1999) 73-112.",
+ "Frank R. Bernhart, \u003ca href=\"/A000296/a000296_1.pdf\"\u003eFundamental chromatic numbers\u003c/a\u003e, Unpublished. (Annotated scanned copy)",
+ "Frank R. Bernhart \u0026 N. J. A. Sloane, \u003ca href=\"/A001683/a001683.pdf\"\u003eCorrespondence, 1977\u003c/a\u003e",
+ "Frank R. Bernhart \u0026 N. J. A. Sloane, \u003ca href=\"/A006343/a006343.pdf\"\u003eEmails, April-May 1994\u003c/a\u003e",
+ "Tom Braden and Artem Vysogorets, \u003ca href=\"https://arxiv.org/abs/1909.09888\"\u003eKazhdan-Lusztig polynomials of matroids under deletion\u003c/a\u003e, arXiv:1909.09888 [math.CO], 2019.",
+ "David Callan, \u003ca href=\"http://www.stat.wisc.edu/~callan/papersother/\"\u003eRiordan numbers are differences of trinomial coefficients\u003c/a\u003e, September 25, 2006.",
+ "D. Callan, \u003ca href=\"http://dx.doi.org/10.1016/j.disc.2008.11.019\"\u003ePattern avoidance in \"flattened\" partitions\u003c/a\u003e, Discrete Math., 309 (2009), 4187-4191.",
+ "David Callan, \u003ca href=\"https://arxiv.org/abs/1702.06150\"\u003eBijections for Dyck paths with all peak heights of the same parity\u003c/a\u003e, arXiv:1702.06150 [math.CO], 2017.",
+ "Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, \u003ca href=\"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8\"\u003eCombinatorial proofs of addition formulas\u003c/a\u003e, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.",
+ "W. Y. C. Chen, E. Y. P. Deng and L. L. M. Yang, \u003ca href=\"https://arxiv.org/abs/math/0602298\"\u003eRiordan paths and derangements\u003c/a\u003e, arXiv:math/0602298 [math.CO], 2006.",
+ "Johann Cigler and Christian Krattenthaler, \u003ca href=\"https://arxiv.org/abs/2003.01676\"\u003eHankel determinants of linear combinations of moments of orthogonal polynomials\u003c/a\u003e, arXiv:2003.01676 [math.CO], 2020.",
+ "Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, \u003ca href=\"http://arxiv.org/abs/1511.06623\"\u003eFrom Entanglement Witness to Generalized Catalan Numbers\u003c/a\u003e, arXiv:1511.06623 [quant-ph], 2015 (see equation (23)).",
+ "Benoit Collins, Ion Nechita, and Deping Ye, \u003ca href=\"http://arxiv.org/abs/1108.1935\"\u003eThe absolute positive partial transpose property for random induced states\u003c/a\u003e, arXiv preprint arXiv:1108.1935 [math-ph], 2011.",
+ "R. De Castro, A. L. Ramírez and J. L. Ramírez, \u003ca href=\"http://arxiv.org/abs/1310.2449\"\u003eApplications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs\u003c/a\u003e, arXiv preprint arXiv:1310.2449 [cs.DM], 2013.",
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+ "S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, \u003ca href=\"http://staff.ru.is/henningu/papers/maps/maps.pdf\"\u003eRestricted non-separable planar maps and some pattern avoiding permutations\u003c/a\u003e, 2012. - From _N. J. A. Sloane_, Jan 01 2013",
+ "D. E. Knuth, \u003ca href=\"/A001006/a001006_3.pdf\"\u003eLetter to L. W. Shapiro, R. K. Guy. N. J. A. Sloane, R. P. Stanley, H. Wilf regarding A001006 and A005043\u003c/a\u003e",
+ "J. W. Layman, \u003ca href=\"http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html\"\u003eThe Hankel Transform and Some of its Properties\u003c/a\u003e, J. Integer Sequences, 4 (2001), #01.1.5.",
+ "Boyu Li, \u003ca href=\"https://uwspace.uwaterloo.ca/bitstream/handle/10012/8179/Boyu_Li.pdf?sequence=1\"\u003eAsymptotic Distributions for Block Statistics on Non-crossing Partitions\u003c/a\u003e, Master's Thesis, Univ. Waterloo, 2013.",
+ "Nanna Holmgaard List, Timothé Romain Léo Melin, Martin van Horn, and Trond Saue, \u003ca href=\"https://arxiv.org/abs/2001.10738\"\u003eBeyond the electric-dipole approximation in simulations of X-ray absorption spectroscopy: Lessons from relativistic theory\u003c/a\u003e, arXiv:2001.10738 [physics.chem-ph], 2020.",
+ "Piera Manara and Claudio Perelli Cippo, \u003ca href=\"http://www.mat.unisi.it/newsito/puma/public_html/22_2/manara_perelli-cippo.pdf\"\u003eThe fine structure of 4321 avoiding involutions and 321 avoiding involutions\u003c/a\u003e, PU. M. A. Vol. 22 (2011), 227-238.",
+ "Peter McCalla and Asamoah Nkwanta, \u003ca href=\"https://arxiv.org/abs/1901.07092\"\u003eCatalan and Motzkin Integral Representations\u003c/a\u003e, arXiv:1901.07092 [math.NT], 2019.",
+ "Zhousheng Mei and Suijie Wang, \u003ca href=\"https://arxiv.org/abs/1804.06265\"\u003ePattern Avoidance of Generalized Permutations\u003c/a\u003e, arXiv:1804.06265 [math.CO], 2018.",
+ "J. Menashe, \u003ca href=\"https://www.whitman.edu/mathematics/SeniorProjectArchive/2007/menashjv.pdf\"\u003eBijections on Riordan objects\u003c/a\u003e [From Tom Copeland, Nov 07 2014]",
+ "D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, \u003ca href=\"http://dx.doi.org/10.4153/CJM-1997-015-x\"\u003eOn some alternative characterizations of Riordan arrays\u003c/a\u003e, Canad. J. Math., 49 (1997), 301-320.",
+ "S. Morrison, E. Peters, and N. Snyder, \u003ca href=\"http://arxiv.org/abs/1501.06869\"\u003eCategories generated by a trivalent vertex\u003c/a\u003e, arXiv preprint arXiv:1501.06869 [math.QA], 2015.",
+ "Jocelyn Quaintance and Harris Kwong, \u003ca href=\"http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html\"\u003eA combinatorial interpretation of the Catalan and Bell number difference tables\u003c/a\u003e, Integers, 13 (2013), #A29.",
+ "J. Riordan, \u003ca href=\"http://dx.doi.org/10.1016/S0097-3165(75)80010-0\"\u003eEnumeration of plane trees by branches and endpoints\u003c/a\u003e, J. Combinat. Theory, Ser A, 19, 214-222, 1975.",
+ "E. Rowland and R. Yassawi, \u003ca href=\"http://arxiv.org/abs/1310.8635\"\u003eAutomatic congruences for diagonals of rational functions\u003c/a\u003e, arXiv preprint arXiv:1310.8635 [math.NT], 2013.",
+ "E. Royer, \u003ca href=\"http://www.carva.org/emmanuel.royer\"\u003eInterpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique\u003c/a\u003e",
+ "E. Royer, \u003ca href=\"http://dx.doi.org/10.1016/S0012-9593(03)00024-7\"\u003eInterprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique\u003c/a\u003e, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620.",
+ "Martin Rubey and Christian Stump, \u003ca href=\"https://arxiv.org/abs/1708.05092\"\u003eDouble deficiencies of Dyck paths via the Billey-Jockusch-Stanley bijection\u003c/a\u003e, arXiv:1708.05092 [math.CO], 2017.",
+ "J. Salas and A. D. Sokal, \u003ca href=\"http://arxiv.org/abs/0711.1738\"\u003eTransfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial\u003c/a\u003e, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738 [math.QA]. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014",
+ "L. W. Shapiro \u0026 N. J. A. Sloane, \u003ca href=\"/A006318/a006318_1.pdf\"\u003eCorrespondence, 1976\u003c/a\u003e",
+ "L. W. Shapiro and C. J. Wang, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL12/Shapiro/shapiro7.html\"\u003eA bijection between 3-Motzkin paths and Schroder paths with no peak at odd height\u003c/a\u003e, JIS 12 (2009) 09.3.2.",
+ "M. Shattuck, \u003ca href=\"http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from93to101.pdf\"\u003eOn the zeros of some polynomials with combinatorial coefficients\u003c/a\u003e, Annales Mathematicae et Informaticae, 42 (2013) pp. 93-101.",
+ "H. C. H. Schubert, \u003ca href=\"http://dx.doi.org/10.1007/BF01446537\"\u003eAllgemeine Anzahlfunctionen für Kegelschnitte, Flächen und Räume zweiten Grades in n Dimensionen\u003c/a\u003e, Math. Annalen, June 1894, Volume 45, Issue 2, pp 153-206.",
+ "Hua Sun and Yi Wang, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL17/Wang/wang11.html\"\u003eA Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers\u003c/a\u003e, J. Int. Seq. 17 (2014) # 14.5.2",
+ "Yidong Sun and Fei Ma, \u003ca href=\"http://arxiv.org/abs/1305.2015\"\u003eMinors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths\u003c/a\u003e, arXiv preprint arXiv:1305.2015 [math.CO], 2013.",
+ "Chao-Jen Wang, \u003ca href=\"http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf\"\u003eApplications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords\u003c/a\u003e.",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/IsotropicTensor.html\"\u003eIsotropic Tensor.\u003c/a\u003e",
+ "F. Yano and H. Yoshida, \u003ca href=\"http://dx.doi.org/10.1016/j.disc.2007.03.050\"\u003eSome set partition statistics in non-crossing partitions and generating functions\u003c/a\u003e, Discr. Math., 307 (2007), 3147-3160."
+ ],
+ "formula": [
+ "a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A000108(k). a(n) = (1/(n+1)) * Sum_{k=0..ceiling(n/2)} binomial(n+1, k)*binomial(n-k-1, k-1), for n \u003e 1. - _Len Smiley_. [Comment from Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 02 2010: the latter sum should be over the range k=1..floor(n/2).]",
+ "G.f.: (1 + x - sqrt(1-2*x-3*x^2))/(2*x*(1+x)).",
+ "G.f.: 2/(1+x+sqrt(1-2*x-3*x^2)). - Paul Peart (ppeart(AT)fac.howard.edu), May 27 2000",
+ "a(n+1) + (-1)^n = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0). - Bernhart",
+ "a(n) = (1/(n+1)) * Sum_{i} (-1)^i*binomial(n+1, i)*binomial(2*n-2*i, n-i). - Bernhart",
+ "G.f. A(x) satisfies A = 1/(1+x) + x*A^2.",
+ "E.g.f.: exp(x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - _Vladeta Jovovic_, Apr 28 2003",
+ "a(n) = A001006(n-1) - a(n-1).",
+ "a(n+1) = Sum_{k=0..n} (-1)^k*A026300(n, k), where A026300 is the Motzkin triangle.",
+ "a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(k, floor(k/2)). - _Paul Barry_, Jan 27 2005",
+ "a(n) = Sum_{k\u003e=0} A086810(n-k, k). - _Philippe Deléham_, May 30 2005",
+ "a(n+2) = Sum_{k\u003e=0} A064189(n-k, k). - _Philippe Deléham_, May 31 2005",
+ "Moment representation: a(n) = (1/(2*Pi))*Int(x^n*sqrt((1+x)(3-x))/(1+x),x,-1,3). - _Paul Barry_, Jul 09 2006",
+ "Inverse binomial transform of A000108 (Catalan numbers). - _Philippe Deléham_, Oct 20 2006",
+ "a(n) = (2/Pi)* Integral_{t_0..Pi} (4*cos^2(x)-1)^n*sin^2(x) dx. - _Andrew V. Sutherland_, Dec 02 2007",
+ "G.f.: 1/(1-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-.....(continued fraction). - _Paul Barry_, Jan 22 2009",
+ "G.f.: 1/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). - _Paul Barry_, May 16 2009",
+ "G.f.: 1/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-... (continued fraction). - _Paul Barry_, Mar 02 2010",
+ "a(n) = -(-1)^n * hypergeom([1/2, n+2],[2],4/3) / sqrt(-3). - _Mark van Hoeij_, Jul 02 2010",
+ "a(n) = (-1)^n*hypergeometric([-n,1/2],[2],4). - _Peter Luschny_, Aug 15 2012",
+ "Let A(x) be the g.f., then x*A(x) is the reversion of x/(1 + x^2*Sum_{k\u003e=0} x^k); see A215340 for the correspondence to Dyck paths without length-1 ascents. - _Joerg Arndt_, Aug 19 2012 and Apr 16 2013",
+ "a(n) ~ 3^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 02 2012",
+ "G.f.: 2/(1+x+1/G(0)), where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 05 2013",
+ "D-finite (an alternative): (n+1)*a(n) = 3*(n-2)*a(n-3) + (5*n-7)*a(n-2) + (n-2)*a(n-1), n \u003e= 3. - _Fung Lam_, Mar 22 2014",
+ "Asymptotics: a(n) = 3^(n+2)/sqrt(3*n*Pi)/(8*n)*(1-21/(16*n) + O(1/n^2)) (with contribution by Vaclav Kotesovec). - _Fung Lam_, Mar 22 2014",
+ "a(n) = T(2*n-1,n)/n, where T(n,k) = triangle of A180177. - _Vladimir Kruchinin_, Sep 23 2014",
+ "a(n) = (-1)^n*JacobiP(n,1,-n-3/2,-7)/(n+1). - _Peter Luschny_, Sep 23 2014",
+ "a(n) = Sum_{k=0..n} C(n,k)*(C(k,n-k)-C(k,n-k-1)). - _Peter Luschny_, Oct 01 2014",
+ "a(n) = A002426(n) - A005717(n), n \u003e 0. - _Mikhail Kurkov_, Feb 24 2019",
+ "a(n) = A309303(n) + A309303(n+1). - _Vladimir Reshetnikov_, Jul 22 2019"
+ ],
+ "example": [
+ "a(5)=6 because the only dissections of a polygon with a total number of 6 edges are: five pentagons with one of the five diagonals and the hexagon with no diagonals.",
+ "G.f. = 1 + x^2 + x^3 + 3*x^4 + 6*x^5 + 15*x^6 + 36*x^7 + 91*x^8 + 232*x^9 + ..."
+ ],
+ "maple": [
+ "A005043 := proc(n) option remember; if n \u003c= 1 then 1-n else (n-1)*(2*A005043(n-1)+3*A005043(n-2))/(n+1); fi; end;",
+ "Order := 20: solve(series((x-x^2)/(1-x+x^2),x)=y,x); # outputs g.f."
+ ],
+ "mathematica": [
+ "a[0]=1; a[1]=0; a[n_]:= a[n] = (n-1)*(2*a[n-1] + 3*a[n-2])/(n+1); Table[ a[n], {n, 0, 30}] (* _Robert G. Wilson v_, Jun 14 2005 *)",
+ "Table[(-3)^(1/2)/6 * (-1)^n*(3*Hypergeometric2F1[1/2,n+1,1,4/3]+ Hypergeometric2F1[1/2,n+2,1,4/3]), {n,0,32}] (* cf. _Mark van Hoeij_ in A001006 *) (* _Wouter Meeussen_, Jan 23 2010 *)",
+ "RecurrenceTable[{a[0]==1,a[1]==0,a[n]==(n-1) (2a[n-1]+3a[n-2])/(n+1)},a,{n,30}] (* _Harvey P. Dale_, Sep 27 2013 *)",
+ "a[ n_]:= SeriesCoefficient[2/(1+x +Sqrt[1-2x-3x^2]), {x, 0, n}]; (* _Michael Somos_, Aug 21 2014 *)",
+ "a[ n_]:= If[n\u003c0, 0, 3^(n+3/2) Hypergeometric2F1[3/2, n+2, 2, 4]/I]; (* _Michael Somos_, Aug 21 2014 *)",
+ "Table[3^(n+3/2) CatalanNumber[n] (4(5+2n)Hypergeometric2F1[3/2, 3/2, 1/2-n, 1/4] -9 Hypergeometric2F1[3/2, 5/2, 1/2 -n, 1/4])/(4^(n+3) (n+1)), {n, 0, 31}] (* _Vladimir Reshetnikov_, Jul 21 2019 *)"
+ ],
+ "program": [
+ "(PARI) {a(n) = if( n\u003c0, 0, n++; polcoeff( serreverse( (x - x^3) / (1 + x^3) + x * O(x^n)), n))}; /* _Michael Somos_, May 31 2005 */",
+ "(Maxima) a[0]:1$",
+ "a[1]:0$",
+ "a[n]:=(n-1)*(2*a[n-1]+3*a[n-2])/(n+1)$",
+ "makelist(a[n],n,0,12); /* _Emanuele Munarini_, Mar 02 2011 */",
+ "(Haskell)",
+ "a005043 n = a005043_list !! n",
+ "a005043_list = 1 : 0 : zipWith div",
+ " (zipWith (*) [1..] (zipWith (+)",
+ " (map (* 2) $ tail a005043_list) (map (* 3) a005043_list))) [3..]",
+ "-- _Reinhard Zumkeller_, Jan 31 2012",
+ "(PARI) N=66; Vec(serreverse(x/(1+x*sum(k=1,N,x^k))+O(x^N))) \\\\ _Joerg Arndt_, Aug 19 2012",
+ "(Sage)",
+ "A005043 = lambda n: (-1)^n*jacobi_P(n,1,-n-3/2,-7)/(n+1)",
+ "[simplify(A005043(n)) for n in (0..29)]",
+ "# _Peter Luschny_, Sep 23 2014",
+ "(Sage)",
+ "def ms():",
+ " a, b, c, d, n = 0, 1, 1, -1, 1",
+ " yield 1",
+ " while True:",
+ " yield -b + (-1)^n*d",
+ " n += 1",
+ " a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)/((n+1)*(n-1))",
+ " c, d = d, (3*(n-1)*c-(2*n-1)*d)/n",
+ "A005043 = ms()",
+ "print([next(A005043) for _ in range(32)]) # _Peter Luschny_, May 16 2016"
+ ],
+ "xref": [
+ "Row sums of triangle A020474, first differences of A082395.",
+ "First diagonal of triangular array in A059346.",
+ "Binomial transform of A126930. - _Philippe Deléham_, Nov 26 2009",
+ "The Hankel transform of a(n+1) is A128834. The Hankel transform of a(n+2) is floor((2*n+4)/3) = A004523(n+2). - _Paul Barry_, Mar 08 2011",
+ "The Kn11 triangle sums of triangle A175136 lead to A005043(n+2), while the Kn12(n) = A005043(n+4)-2^(n+1), Kn13(n) = A005043(n+6)-(n^2+9*n+56)*2^(n-2) and the Kn4(n) = A005043(2*n+2) = A099251(n+1) triangle sums are related to the sequence given above. For the definitions of these triangle sums see A180662. - _Johannes W. Meijer_, May 06 2011",
+ "The self-convolution of A005043 gives A187306. - _Philippe Deléham_, Jan 28 2014",
+ "Cf. A000045, A000108, A000957, A001006, A007317, A057078, A091867, A104597, A126120, A178514, A249548, A309303.",
+ "Bisections: A099251, A099252."
+ ],
+ "keyword": "nonn,easy,nice,changed",
+ "offset": "0,5",
+ "author": "_N. J. A. Sloane_",
+ "ext": [
+ "Thanks to Laura L. M. Yang (yanglm(AT)hotmail.com) for a correction, Aug 29 2004",
+ "Name changed to Riordan numbers following a suggestion from _Ira M. Gessel_. - _N. J. A. Sloane_, Jul 24 2020"
+ ],
+ "references": 130,
+ "revision": 412,
+ "time": "2021-04-15T05:26:09-04:00",
+ "created": "1991-07-11T03:00:00-04:00"
+ },
+ {
+ "number": 99323,
+ "data": "1,1,0,1,-1,3,-6,15,-36,91,-232,603,-1585,4213,-11298,30537,-83097,227475,-625992,1730787,-4805595,13393689,-37458330,105089229,-295673994,834086421,-2358641376,6684761125,-18985057351,54022715451,-154000562758,439742222071,-1257643249140",
+ "name": "Expansion of (sqrt(1+3*x) + sqrt(1-x))/(2*sqrt(1-x)).",
+ "comment": [
+ "Binomial transform is A072100. Signed Motzkin numbers with an additional leading 1.",
+ "Inverse binomial transform of A001405 gives this without the initial 1. So does the binomial transform of (-1)^n*A000108(n) = [1,-1,2,-5,14,-42,...]. - _Philippe Deléham_, Mar 20 2007"
+ ],
+ "link": [
+ "C. Banderier and D. Merlini, \u003ca href=\"http://algo.inria.fr/banderier/Papers/infjumps.ps\"\u003eLattice paths with an infinite set of jumps\u003c/a\u003e, FPSAC'02 Melbourne, 2002."
+ ],
+ "formula": [
+ "a(n) = 0^n + Sum_{k=0..n-1} binomial(n-1,k)*(-1)^k*C(k), where C(k) is the k-th Catalan number.",
+ "G.f.: 1 + x/(1-sqrt(x))/G(0), where G(k)= 1 + sqrt(x)/(1 - sqrt(x)/(1 + x/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 28 2013",
+ "D-finite with recurrence: n*a(n) + 2*(n-2)*a(n-1) + 3*(-n+2)*a(n-2) = 0. - _R. J. Mathar_, Oct 10 2014",
+ "a(n) ~ -(-1)^n * 3^(n + 1/2) / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 31 2017"
+ ],
+ "maple": [
+ "with(PolynomialTools): CoefficientList(convert(taylor((sqrt(1 + 3*x) + sqrt(1 - x))/2/sqrt(1 - x), x = 0, 33), polynom), x); # _Taras Goy_, Aug 07 2017"
+ ],
+ "mathematica": [
+ "CoefficientList[Series[(Sqrt[1+3x]+Sqrt[1-x])/(2Sqrt[1-x]),{x,0,40}],x] (* _Harvey P. Dale_, Feb 06 2015 *)"
+ ],
+ "xref": [
+ "Cf. A000108, A005043."
+ ],
+ "keyword": "easy,sign",
+ "offset": "0,6",
+ "author": "_Paul Barry_, Oct 12 2004",
+ "ext": [
+ "Edited by _N. J. A. Sloane_, Oct 05 2009"
+ ],
+ "references": 15,
+ "revision": 35,
+ "time": "2020-01-30T21:29:15-05:00",
+ "created": "2005-02-20T03:00:00-05:00"
+ },
+ {
+ "number": 174297,
+ "data": "1,-1,-1,0,-1,1,-3,6,-15,36,-91,232,-603,1585,-4213,11298,-30537,83097,-227475,625992,-1730787,4805595,-13393689,37458330,-105089229,295673994,-834086421,2358641376,-6684761125,18985057351,-54022715451",
+ "name": "First column of A174295.",
+ "comment": [
+ "First 6 terms as in Mobius function A008683. Signed version of A099323 with an additional leading 1."
+ ],
+ "formula": [
+ "a(n) = -(-3)^(n-3/2)*hypergeom([3/2, n-1],[2],4) for n \u003e 2 (guessed formula) [From _Mark van Hoeij_, Jul 02 2010]"
+ ],
+ "xref": [
+ "Cf. A112468, A112467, A174294, A174295, A174296, A174297."
+ ],
+ "keyword": "sign",
+ "offset": "1,7",
+ "author": "_Mats Granvik_, Mar 15 2010",
+ "references": 6,
+ "revision": 4,
+ "time": "2012-03-31T10:24:32-04:00",
+ "created": "2010-06-01T03:00:00-04:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,2,3,4,5,6,.json b/data/1,2,3,4,5,6,.json
new file mode 100644
index 0000000..f6e3372
--- /dev/null
+++ b/data/1,2,3,4,5,6,.json
@@ -0,0 +1,957 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,2,3,4,5,6,7,8,9,10,",
+ "count": 1807,
+ "start": 0,
+ "results": [
+ {
+ "number": 27,
+ "id": "M0472 N0173",
+ "data": "1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77",
+ "name": "The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.",
+ "comment": [
+ "For some authors, the terms \"natural numbers\" and \"counting numbers\" include 0, i.e., refer to the nonnegative integers A001477; the term \"whole numbers\" frequently also designates the whole set of (signed) integers A001057.",
+ "a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).",
+ "Inverse Euler transform of A000219.",
+ "The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - _Clark Kimberling_, Apr 05 2003",
+ "For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - _Clark Kimberling_, Jan 09 2005",
+ "Numbers of form (2^i)*k for odd k (i.e., n = A006519(n)*A000265(n)); thus n corresponds uniquely to an ordered pair (i,k) where i=A007814, k=A000265 (with A007814(2n)=A001511(n), A007814(2n+1)=0). - _Lekraj Beedassy_, Apr 22 2006",
+ "If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points), A000124 (number of regions in 2-space defined by n straight lines), A000125 (number of regions in 3-space defined by n planes), A000127 (number of regions in 4-space defined by n hyperplanes), A006261, A008859, A008860, A008861, A008862 and A008863, where the last six sequences are interpreted analogously and in each \"... by n ...\" clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006",
+ "Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments on A000124 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006",
+ "The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. - _James East_, May 03 2007",
+ "The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. - _James East_, May 03 2007",
+ "\"God made the integers; all else is the work of man.\" This famous quotation is a translation of \"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk,\" spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's \"Leopold Kronecker,\" Jahresberichte D.M.V. 2 (1893) 5-31. - _Clark Kimberling_, Jul 07 2007",
+ "Binomial transform of A019590, inverse binomial transform of A001792. - _Philippe Deléham_, Oct 24 2007",
+ "Writing A000027 as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. - _Clark Kimberling_, Sep 11 2008",
+ "A000007(a(n)) = 0; A057427(a(n)) = 1. - _Reinhard Zumkeller_, Oct 12 2008",
+ "a(n) is also the mean of the first n odd integers. - _Ian Kent_, Dec 23 2008",
+ "Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - _Gary W. Adamson_, Jun 05 2009",
+ "These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - _Michael B. Porter_, Oct 08 2009",
+ "Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. - _Jaroslav Krizek_, Oct 18 2009",
+ "Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1\u003c=j\u003c=k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) \u003c n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)=A000217(n), the n-th triangular number. - _Dennis P. Walsh_, Nov 19 2009",
+ "Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. - _Jaroslav Krizek_, Dec 11 2009",
+ "a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - _Leonid Bedratyuk_, Jan 04 2010",
+ "Floyd's triangle read by rows. - _Paul Muljadi_, Jan 25 2010",
+ "Number of numbers between k and 2k where k is an integer. - _Giovanni Teofilatto_, Mar 26 2010",
+ "Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown in A178568. - _Gary W. Adamson_, May 29 2010",
+ "1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. - _Gary W. Adamson_, Jul 15 2010",
+ "Number of n-digit numbers the binary expansion of which contains one run of 1's. - _Vladimir Shevelev_, Jul 30 2010",
+ "From _Clark Kimberling_, Jan 29 2011: (Start)",
+ "Let T denote the \"natural number array A000027\":",
+ " 1 2 4 7 ...",
+ " 3 5 8 12 ...",
+ " 6 9 13 18 ...",
+ " 10 14 19 25 ...",
+ "T(n,k) = n+(n+k-2)*(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End)",
+ "The Stern polynomial B(n,x) evaluated at x=2. See A125184. - _T. D. Noe_, Feb 28 2011",
+ "The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - _Mohammad K. Azarian_, Oct 13 2011",
+ "As a function of Bernoulli numbers B_n (cf. A027641: (1, -1/2, 1/6, 0, -1/30, 0, 1/42, ...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangle A074909 (the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30, ...] = the vector [1, 2, 3, 4, 5, ...]. - _Gary W. Adamson_, Mar 05 2012",
+ "Number of partitions of 2n+1 into exactly two parts. - _Wesley Ivan Hurt_, Jul 15 2013",
+ "Integers n dividing u(n) = 2u(n-1) - u(n-2); u(0)=0, u(1)=1 (Lucas sequence A001477). - _Thomas M. Bridge_, Nov 03 2013",
+ "For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) = A194807. - _Stanislav Sykora_, Jan 20 2014",
+ "Engel expansion of e-1 (A091131 = 1.71828...). - _Jaroslav Krizek_, Jan 23 2014",
+ "a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - _Manda Riehl_, Aug 05 2014",
+ "a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - _Manda Riehl_ Aug 07 2014",
+ "a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) \u003c 1/n. - _Clark Kimberling_, Sep 28 2014",
+ "a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 \u003c 1/n. - _Clark Kimberling_, Oct 02 2014",
+ "Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). - _Ryan Stees_, Dec 15 2014",
+ "As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...}, A001477, to the positive integers {1,2,3,...}. - _M. F. Hasler_, Jan 18 2015",
+ "See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i\u003e=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - _Alexander R. Povolotsky_, Apr 24 2015",
+ "a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016",
+ "Does not satisfy Benford's law [Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017",
+ "Parametrization for the finite multisubsets of the positive integers, where, for p_j the j-th prime, n = Prod_j p_j^{e_j} corresponds to the multiset containing e_j copies of j ('Heinz encoding' -- see A056239, A003963, A289506, A289507, A289508, A289509) - _Christopher J. Smyth_, Jul 31 2017",
+ "The arithmetic function v_1(n,1) as defined in A289197. - _Robert Price_, Aug 22 2017",
+ "For n\u003e=3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. - _Michel Marcus_, Apr 28 2018",
+ "a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. - _Nick Mayers_, Jun 08 2018",
+ "Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. - _Frank Hollstein_, Mar 25 2019",
+ "(1, 2, 3, 4, 5,...) is the fourth INVERT transform of (1, -2, 3, -4, 5,...). - Gary W. Adamson_, Jul 15 2019"
+ ],
+ "reference": [
+ "T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.",
+ "T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.",
+ "W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From _Leonid Bedratyuk_, Jan 04 2010]",
+ "I. S. Gradstein and I. M. Ryshik, Tables of series, products , and integrals, Volume 1, Verlag Harri Deutsch, 1981.",
+ "R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.",
+ "N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).",
+ "N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence)."
+ ],
+ "link": [
+ "N. J. A. Sloane, \u003ca href=\"/A000027/b000027.txt\"\u003eTable of n, a(n) for n = 1..500000\u003c/a\u003e [a large file]",
+ "Archimedes Laboratory, \u003ca href=\"http://www.archimedes-lab.org/numbers/Num1_200.html\"\u003eWhat's special about this number?\u003c/a\u003e",
+ "Affaire de Logique, \u003ca href=\"http://www.affairedelogique.com/espace_probleme.php?corps=probleme\u0026amp;num=1051\"\u003ePick et Pick et Colegram\u003c/a\u003e (in French), No. 1051, 18-04-2018.",
+ "Paul Barry, \u003ca href=\"http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html\"\u003eA Catalan Transform and Related Transformations on Integer Sequences\u003c/a\u003e, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.",
+ "James Barton, \u003ca href=\"http://www.virtuescience.com/number-list.html\"\u003eThe Numbers\u003c/a\u003e",
+ "A. Berger and T. P. Hill, \u003ca href=\"http://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf\"\u003eWhat is Benford's Law?\u003c/a\u003e, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.",
+ "A. Breiland, L. Oesper, and L. Taalman, \u003ca href=\"http://educ.jmu.edu/~taalmala/breil_oesp_taal.pdf\"\u003ep-Coloring classes of torus knots\u003c/a\u003e, Online Missouri J. Math. Sci., 21 (2009), 120-126.",
+ "N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, \u003ca href=\"http://projecteuclid.org/euclid.mjms/1312232716\"\u003eSpiral knots\u003c/a\u003e, Missouri J. of Math. Sci., 22 (2010).",
+ "C. K. Caldwell, \u003ca href=\"http://primes.utm.edu/curios\"\u003ePrime Curios\u003c/a\u003e",
+ "Case and Abiessu, \u003ca href=\"http://everything2.net/index.pl?node_id=17633\u0026amp;displaytype=printable\u0026amp;lastnode_id=17633\"\u003einteresting number\u003c/a\u003e",
+ "S. Crandall, \u003ca href=\"http://tingilinde.typepad.com/starstuff/2005/11/significant_int.html\"\u003enotes on interesting digital ephemera\u003c/a\u003e",
+ "O. Curtis, \u003ca href=\"http://users.pipeline.com.au/owen/Numbers.html\"\u003eInteresting Numbers\u003c/a\u003e",
+ "M. DeLong, M. Russell, and J. Schrock, \u003ca href=\"http://dx.doi.org/10.2140/involve.2015.8.361\"\u003eColorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)\u003c/a\u003e, Involve, Vol. 8 (2015), No. 3, 361-384.",
+ "Walter Felscher, \u003ca href=\"http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may99/0210.html\"\u003eHistoria Matematica Mailing List Archive.\u003c/a\u003e",
+ "P. Flajolet and R. Sedgewick, \u003ca href=\"http://algo.inria.fr/flajolet/Publications/books.html\"\u003eAnalytic Combinatorics\u003c/a\u003e, 2009; see page 371",
+ "Robert R. Forslund, \u003ca href=\"http://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps\"\u003eA logical alternative to the existing positional number system\u003c/a\u003e, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.",
+ "E. Friedman, \u003ca href=\"https://erich-friedman.github.io/numbers.html\"\u003eWhat's Special About This Number?\u003c/a\u003e",
+ "R. K. Guy, \u003ca href=\"/A000346/a000346.pdf\"\u003eLetter to N. J. A. Sloane\u003c/a\u003e",
+ "Milan Janjic, \u003ca href=\"http://www.pmfbl.org/janjic/\"\u003eEnumerative Formulas for Some Functions on Finite Sets\u003c/a\u003e",
+ "Kival Ngaokrajang, \u003ca href=\"/A000027/a000027_2.pdf\"\u003eIllustration about relation to many other sequences\u003c/a\u003e, when the sequence is considered as a triangular table read by its antidiagonals. \u003ca href=\"/A000027/a000027_3.pdf\"\u003eAdditional illustrations\u003c/a\u003e when the sequence is considered as a centered triangular table read by rows.",
+ "M. Keith, \u003ca href=\"http://users.aol.com/s6sj7gt/interest.htm\"\u003eAll Numbers Are Interesting: A Constructive Approach\u003c/a\u003e",
+ "Leonardo of Pisa [Leonardo Pisano], \u003ca href=\"/A000027/a000027.jpg\"\u003eIllustration of initial terms\u003c/a\u003e, from Liber Abaci [The Book of Calculation], 1202 (photo by David Singmaster).",
+ "R. Munafo, \u003ca href=\"http://www.mrob.com/pub/math/numbers.html\"\u003eNotable Properties of Specific Numbers\u003c/a\u003e",
+ "G. Pfeiffer, \u003ca href=\"http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html\"\u003eCounting Transitive Relations\u003c/a\u003e, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.",
+ "R. Phillips, \u003ca href=\"http://richardphillips.org.uk/number/Num1.htm\"\u003eNumbers from one to thirty-one\u003c/a\u003e",
+ "J. Striker, \u003ca href=\"http://www.ams.org/publications/journals/notices/201706/rnoti-p543.pdf\"\u003eDynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance\u003c/a\u003e, Notices of the AMS, June/July 2017, pp. 543-549.",
+ "G. Villemin's Almanac of Numbers, \u003ca href=\"http://villemin.gerard.free.fr/aNombre/Nb0a50.htm\"\u003eNOMBRES en BREF (in French)\u003c/a\u003e",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/NaturalNumber.html\"\u003eNatural Number\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/PositiveInteger.html\"\u003ePositive Integer\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/CountingNumber.html\"\u003eCounting Number\u003c/a\u003e \u003ca href=\"http://mathworld.wolfram.com/Composition.html\"\u003eComposition\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/Davenport-SchinzelSequence.html\"\u003eDavenport-Schinzel Sequence\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/IdempotentNumber.html\"\u003eIdempotent Number\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/N.html\"\u003eN\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/SmarandacheCeilFunction.html\"\u003eSmarandache Ceil Function\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/WholeNumber.html\"\u003eWhole Number\u003c/a\u003e, \u003ca href=\"http://mathworld.wolfram.com/EngelExpansion.html\"\u003eEngel Expansion\u003c/a\u003e, and \u003ca href=\"http://mathworld.wolfram.com/TrinomialCoefficient.html\"\u003eTrinomial Coefficient\u003c/a\u003e",
+ "Wikipedia, \u003ca href=\"http://en.wikipedia.org/wiki/List_of_numbers\"\u003eList of numbers\u003c/a\u003e, \u003ca href=\"http://en.wikipedia.org/wiki/Interesting_number_paradox\"\u003eInteresting number paradox\u003c/a\u003e, and \u003ca href=\"http://en.wikipedia.org/wiki/Floyd%27s_triangle\"\u003eFloyd's triangle\u003c/a\u003e",
+ "Robert G. Wilson v, \u003ca href=\"/A000027/a000027.txt\"\u003eEnglish names for the numbers from 0 to 11159 without spaces or hyphens \u003c/a\u003e",
+ "Robert G. Wilson v, \u003ca href=\"/A001477/a001477.txt\"\u003eAmerican English names for the numbers from 0 to 100999 without spaces or hyphens\u003c/a\u003e",
+ "\u003ca href=\"/index/Cor#core\"\u003eIndex entries for \"core\" sequences\u003c/a\u003e",
+ "\u003ca href=\"/index/Aa#aan\"\u003eIndex entries for sequences of the a(a(n)) = 2n family\u003c/a\u003e",
+ "\u003ca href=\"/index/Per#IntegerPermutation\"\u003eIndex entries for sequences that are permutations of the natural numbers\u003c/a\u003e",
+ "\u003ca href=\"/index/Par#partN\"\u003eIndex entries for related partition-counting sequences\u003c/a\u003e",
+ "\u003ca href=\"/index/Rec#order_02\"\u003eIndex entries for linear recurrences with constant coefficients\u003c/a\u003e, signature (2,-1).",
+ "\u003ca href=\"/index/Di#divseq\"\u003eIndex to divisibility sequences\u003c/a\u003e",
+ "\u003ca href=\"/index/Be#Benford\"\u003eIndex entries for sequences related to Benford's law\u003c/a\u003e"
+ ],
+ "formula": [
+ "a(2k+1) = A005408(k), k \u003e= 0, a(2k) = A005843(k), k \u003e= 1.",
+ "Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001",
+ "Another g.f.: Sum_{n\u003e0} phi(n)*x^n/(1-x^n) (Apostol).",
+ "When seen as an array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n*(n+1)+1 (A001844), antidiagonal sums are n*(n^2+1)/2 (A006003). - _Ralf Stephan_, Oct 17 2004",
+ "Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005",
+ "G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).",
+ "Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is the g.f. of A000108. - _Michael Somos_, Sep 04 2006",
+ "G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 4*u*v. - _Michael Somos_, Oct 03 2006",
+ "Convolution of A000012 (the all-ones sequence) with itself. - _Tanya Khovanova_, Jun 22 2007",
+ "a(n) = 2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n)=1+a(n-1). - _Philippe Deléham_, Nov 03 2008",
+ "a(n) = A000720(A000040(n)). - _Juri-Stepan Gerasimov_, Nov 29 2009",
+ "a(n+1) = Sum_{k=0..n} A101950(n,k). - _Philippe Deléham_, Feb 10 2012",
+ "a(n) = Sum_{d | n} phi(d) = Sum_{d | n} A000010(d). - _Jaroslav Krizek_, Apr 20 2012",
+ "G.f.: x * Product_{j\u003e=0} (1+x^(2^j))^2 = x * (1+2*x+x^2) * (1+2*x^2+x^4) * (1+2*x^4+x^8) * ... = x + 2x^2 + 3x^3 + ... . - _Gary W. Adamson_, Jun 26 2012",
+ "a(n) = det(binomial(i+1,j), 1 \u003c= i,j \u003c= n). - _Mircea Merca_, Apr 06 2013",
+ "E.g.f.: x*E(0), where E(k)= 1 + 1/(x - x^3/(x^2 + (k+1)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 03 2013",
+ "From _Wolfdieter Lang_, Oct 09 2013: (Start)",
+ "a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n \u003e 1.",
+ "a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n \u003e 1.",
+ "These identities are used in the calculation of products of ratios of lengths of certain lines in a regular n-gon. For the first identity see the Gradstein-Ryshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x -\u003e 0 (L'Hôpital). The second line follows from the first one. Thanks to _Seppo Mustonen_ who led me to consider n-gon lengths products. (End)",
+ "a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k\u003e=0. - _Mircea Merca_, Jan 25 2014",
+ "a(n) = A052410(n)^A052409(n). - _Reinhard Zumkeller_, Apr 06 2014",
+ "a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - _Pierre CAMI_, Apr 25 2014",
+ "a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - _Clark Kimberling_, Oct 08 2014",
+ "a(n) = floor(1/(log(n+1)-log(n))). - _Thomas Ordowski_, Oct 10 2014",
+ "a(k) = det(S(2,k,1)). - _Ryan Stees_, Dec 15 2014",
+ "a(n) = 1/(1/(n+1)+1/(n+1)^2+1/(n+1)^3+.... - _Pierre CAMI_, Jan 22 2015",
+ "a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n \u003e= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m \u003e= 0, from the fact that Stirling2*Stirling1 = identity matrix. See A048993, A048994 and A000110. - _Wolfdieter Lang_, Feb 03 2015",
+ "a(n) = Sum_{k=1...2n-1}(-1)^(k+1)*k*(2n-k). In addition, surprisingly, a(n) = Sum_{k=1...2n-1}(-1)^(k+1)*k^2*(2n-k)^2. - _Charlie Marion_, Jan 05 2016",
+ "G.f.: x/(1-x)^2 = (x * r(x) *r(x^3) * r(x^9) * r(x^27) *...), where r(x) = (1 + x + x^2)^2 = (1 + 2x + 3x^2 + 2x^3 + x^4). - _Gary W. Adamson_, Jan 11 2017",
+ "a(n) = floor(1/(Pi/2-arctan(n))). - _Clark Kimberling_, Mar 11 2020",
+ "a(n) = Sum_{d|n} mu(n/d)*sigma(d). - _Ridouane Oudra_, Oct 03 2020"
+ ],
+ "maple": [
+ "A000027 := n-\u003en; seq(A000027(n), n=1..100);"
+ ],
+ "mathematica": [
+ "Range@ 77 (* _Robert G. Wilson v_, Mar 31 2015 *)"
+ ],
+ "program": [
+ "(MAGMA) [ n : n in [1..100]];",
+ "(PARI) {a(n) = n};",
+ "(R) 1:100",
+ "(Shell) seq 1 100",
+ "(Haskell)",
+ "a000027 = id",
+ "a000027_list = [1..] -- _Reinhard Zumkeller_, May 07 2012",
+ "(Maxima) makelist(n, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */"
+ ],
+ "xref": [
+ "A001477 = nonnegative numbers.",
+ "Partial sums of A000012.",
+ "Cf. A001478, A001906, A007931, A007932, A027641, A074909, A089353 (multisets), A178568, A194807.",
+ "Cf. A026081 = integers in reverse alphabetical order in U.S. English, A107322 = English name for number and its reverse have the same number of letters, A119796 = zero through ten in alphabetical order of English reverse spelling, A005589, etc. Cf. A185787 (includes a list of sequences based on the natural number array A000027).",
+ "Cf. Boustrophedon transforms: A000737, A231179;",
+ "Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).",
+ "Cf. A048993, A048994, A000110 (see the Feb 03 2015 formula).",
+ "Cf. A289187,"
+ ],
+ "keyword": "core,nonn,easy,mult,tabl",
+ "offset": "1,2",
+ "author": "_N. J. A. Sloane_",
+ "ext": [
+ "Links edited by _Daniel Forgues_, Oct 07 2009"
+ ],
+ "references": 1823,
+ "revision": 558,
+ "time": "2020-11-28T11:01:13-05:00",
+ "created": "1991-04-30T03:00:00-04:00"
+ },
+ {
+ "number": 7953,
+ "data": "0,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10,11,3,4,5,6,7,8,9,10,11,12,4,5,6,7,8,9,10,11,12,13,5,6,7,8,9,10,11,12,13,14,6,7,8,9,10,11,12,13,14,15,7,8,9,10,11,12,13,14,15,16,8,9,10,11,12,13,14,15",
+ "name": "Digital sum (i.e., sum of digits) of n; also called digsum(n).",
+ "comment": [
+ "Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).",
+ "Also the fixed point of the morphism 0 -\u003e {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -\u003e {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -\u003e {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - _Robert G. Wilson v_, Jul 27 2006",
+ "For n \u003c 100 equal to (floor(n/10) + n mod 10) = A076314(n). - _Hieronymus Fischer_, Jun 17 2007",
+ "a(n) = A138530(n, 10) for n \u003e 9. - _Reinhard Zumkeller_, Mar 26 2008",
+ "a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - _Reinhard Zumkeller_, Apr 25 2009",
+ "a(n) mod 2 = A179081(n). - _Reinhard Zumkeller_, Jun 28 2010"
+ ],
+ "reference": [
+ "Krassimir Atanassov, On the 16th Smarandache Problem, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 1, 36-38."
+ ],
+ "link": [
+ "N. J. A. Sloane, \u003ca href=\"/A007953/b007953.txt\"\u003eTable of n, a(n) for n = 0..10000\u003c/a\u003e",
+ "Krassimir Atanassov, \u003ca href=\"http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf\"\u003eOn Some of Smarandache's Problems\u003c/a\u003e",
+ "Jean-Luc Baril, \u003ca href=\"http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178\"\u003eClassical sequences revisited with permutations avoiding dotted pattern\u003c/a\u003e, Electronic Journal of Combinatorics, 18 (2011), #P178.",
+ "Ernesto Estrada and Puri Pereira-Ramos, \u003ca href=\"https://doi.org/10.1155/2018/9893867\"\u003eSpatial 'Artistic' Networks: From Deconstructing Integer-Functions to Visual Arts\u003c/a\u003e, Complexity, Vol. 2018 (2018), Article ID 9893867.",
+ "A. O. Gel'fond, \u003ca href=\"http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa13115.pdf\"\u003eSur les nombres qui ont des propriétés additives et multiplicatives données\u003c/a\u003e (French) Acta Arith. 13 1967/1968 259--265. MR0220693 (36 #3745)",
+ "Christian Mauduit and András Sárközy, \u003ca href=\"http://dx.doi.org/10.1006/jnth.1998.2229\"\u003eOn the arithmetic structure of sets characterized by sum of digits properties\u003c/a\u003e J. Number Theory 61(1996), no. 1, 25--38. MR1418316 (97g:11107)",
+ "Christian Mauduit and András Sárközy, \u003ca href=\"http://matwbn.icm.edu.pl/ksiazki/aa/aa81/aa8122.pdf\"\u003e On the arithmetic structure of the integers whose sum of digits is fixed\u003c/a\u003e, Acta Arith. 81 (1997), no. 2, 145--173. MR1456239 (99a:11096)",
+ "Kerry Mitchell, \u003ca href=\"http://kerrymitchellart.com/articles/Spirolateral-Type_Images_from_Integer_Sequences.pdf\"\u003eSpirolateral-Type Images from Integer Sequences\u003c/a\u003e, 2013",
+ "Kerry Mitchell, \u003ca href=\"/A007953/a007953.jpg\"\u003eSpirolateral image for this sequence\u003c/a\u003e [taken, with permission, from the Spirolateral-Type Images from Integer Sequences article]",
+ "Jan-Christoph Puchta and Jürgen Spilker, \u003ca href=\"http://dx.doi.org/10.1007/s00591-002-0048-4\"\u003eAltes und Neues zur Quersumme\u003c/a\u003e, Mathematische Semesterberichte, 49 (2002), 209-226.",
+ "Jan-Christoph Puchta and Jürgen Spilker, \u003ca href=\"http://www.math.uni-rostock.de/~schlage-puchta/papers/Quersumme.pdf\"\u003eAltes und Neues zur Quersumme\u003c/a\u003e",
+ "Maxwell Schneider and Robert Schneider, \u003ca href=\"https://arxiv.org/abs/1807.06710\"\u003eDigit sums and generating functions\u003c/a\u003e, arXiv:1807.06710 [math.NT], 2018.",
+ "Vladimir Shevelev, \u003ca href=\"http://journals.impan.pl/aa/Inf/126-3-1.html\"\u003eCompact integers and factorials\u003c/a\u003e, Acta Arith. 126 (2007), no.3,195-236 (cf. pp.205-206).",
+ "Robert Walker, \u003ca href=\"http://robertinventor.com/ftswiki/Self_Similar_Sloth_Canon_Number_Sequences\"\u003eSelf Similar Sloth Canon Number Sequences\u003c/a\u003e",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/DigitSum.html\"\u003eDigit Sum\u003c/a\u003e",
+ "Wikipedia, \u003ca href=\"http://en.wikipedia.org/wiki/Digit_sum\"\u003eDigit sum\u003c/a\u003e",
+ "\u003ca href=\"/index/Coi#Colombian\"\u003eIndex entries for Colombian or self numbers and related sequences\u003c/a\u003e"
+ ],
+ "formula": [
+ "a(n) \u003c= 9(log_10(n)+1). - _Stefan Steinerberger_, Mar 24 2006",
+ "a(0) = 0, a(10n+i) = a(n) + i 0 \u003c= i \u003c= 9; a(n) = n - 9*(sum(k \u003e 0, floor(n/10^k)) = n - 9*A054899(n). - _Benoit Cloitre_, Dec 19 2002",
+ "From _Hieronymus Fischer_, Jun 17 2007: (Start)",
+ "G.f. g(x) = sum{k \u003e 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).",
+ "a(n) = n - 9*sum{10 \u003c= k \u003c= n, sum{j|k, j \u003e= 10, floor(log_10(j)) - floor(log_10(j-1))}}. (End)",
+ "From _Hieronymus Fischer_, Jun 25 2007: (Start)",
+ "The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k \u003e= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k \u003e 1 is a power of 10, else b(k) = 0.",
+ "G.f.: g(x) = sum{k \u003e 0, (1 - 9*c(k))*x^k}/(1-x), where c(k) = sum{j \u003e 1, j|k, floor(log_10(j)) - floor(log_10(j-1))}.",
+ "a(n) = n - 9*sum_{0 \u003c k \u003c= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)",
+ "From _Hieronymus Fischer_, Oct 06 2007: (Start)",
+ "a(n) \u003c= 9*(1 + floor(log_10(n)), equality holds for n = 10^m - 1, m \u003e 0.",
+ "lim sup (a(n) - 9*log_10(n)) = 0 for n --\u003e infinity.",
+ "lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n --\u003e infinity. (End)",
+ "a(A051885(n)) = n.",
+ "a(n) \u003c= 9*log_10(n+1). - _Vladimir Shevelev_, Jun 01 2011",
+ "a(n) = a(n-1) + a(n-10) - a(n-11), for n \u003c 100. - _Alexander R. Povolotsky_, Oct 09 2011",
+ "a(n) = Sum_k \u003e= 0 {A031298(n, k)}. - _Philippe Deléham_, Oct 21 2011",
+ "a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k \u003e= 0. - _Hieronymus Fischer_, Mar 24 2014"
+ ],
+ "example": [
+ "a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29."
+ ],
+ "maple": [
+ "A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # _R. J. Mathar_, Mar 17 2011"
+ ],
+ "mathematica": [
+ "Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* _Stefan Steinerberger_, Mar 24 2006 *)",
+ "Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)",
+ "Nest[Flatten[# /. a_Integer -\u003e Array[a + # \u0026, 10, 0]] \u0026, {0}, 2] (* _Robert G. Wilson v_, Jul 27 2006 *)",
+ "Table[Sum[Floor[n/10^k] - 10 * Floor[n/10^(k + 1)], {k, 0, Floor[Log[10, n]]}], {n, 300}] (* _José de Jesús Camacho Medina_, Mar 31 2014 *)",
+ "Total/@IntegerDigits[Range[0,90]] (* _Harvey P. Dale_, May 10 2016 *)"
+ ],
+ "program": [
+ "/* The next few PARI programs are kept for historical and pedagogical reasons.",
+ " For practical use, the suggested and most efficient code is: A007953=sumdigits */",
+ "(PARI) a(n)=if(n\u003c1, 0, if(n%10, a(n-1)+1, a(n/10))) \\\\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n\u003e9, n=divrem(n, 10); n[2]+a(n[1]), n)",
+ "(PARI) a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]\u003e=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\\\ _M. F. Hasler_, Mar 22 2011",
+ "(PARI) a(n)=sum(i=1, #n=digits(n), n[i]) \\\\ Twice as fast. Not so nice but faster:",
+ "(PARI) a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\\\ - _M. F. Hasler_, May 10 2015",
+ "/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by _M. F. Hasler_, Nov 09 2018] */",
+ "(PARI) a(n) = sumdigits(n); \\\\ _Altug Alkan_, Apr 19 2018",
+ "(Haskell)",
+ "a007953 n | n \u003c 10 = n",
+ " | otherwise = a007953 n' + r where (n',r) = divMod n 10",
+ "-- _Reinhard Zumkeller_, Nov 04 2011, Mar 19 2011",
+ "(MAGMA) [ \u0026+Intseq(n): n in [0..87] ]; // _Bruno Berselli_, May 26 2011",
+ "(Smalltalk)",
+ "\"Recursive version for general bases. Set base = 10 for this sequence.\"",
+ "digitalSum: base",
+ "| s |",
+ "base = 1 ifTrue: [^self].",
+ "(s := self // base) \u003e 0",
+ " ifTrue: [^(s digitalSum: base) + self - (s * base)]",
+ " ifFalse: [^self]",
+ "\"by _Hieronymus Fischer_, Mar 24 2014\"",
+ "(Python)",
+ "def A007953(n):",
+ " return sum(int(d) for d in str(n)) # _Chai Wah Wu_, Sep 03 2014",
+ "(Scala) (0 to 99).map(_.toString.map(_.toInt - 48).sum) // _Alonso del Arte_, Sep 15 2019",
+ "(Swift 5)",
+ "A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // _Egor Khmara_, Jun 15 2021"
+ ],
+ "xref": [
+ "Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).",
+ "For n + digsum(n) see A062028."
+ ],
+ "keyword": "nonn,base,nice,easy,look",
+ "offset": "0,3",
+ "author": "R. Muller",
+ "ext": [
+ "More terms from _Hieronymus Fischer_, Jun 17 2007",
+ "Edited by _Michel Marcus_, Nov 11 2013"
+ ],
+ "references": 948,
+ "revision": 239,
+ "time": "2021-02-25T02:39:48-05:00",
+ "created": "1996-03-15T03:00:00-05:00"
+ },
+ {
+ "number": 1477,
+ "data": "0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77",
+ "name": "The nonnegative integers.",
+ "comment": [
+ "Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - _N. J. A. Sloane_, Mar 13 2010",
+ "The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - _N. J. A. Sloane_, Jun 16 2010",
+ "a(n) = A007966(n)*A007967(n). - _Reinhard Zumkeller_, Jun 18 2011",
+ "Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - _M. F. Hasler_, Nov 20 2013",
+ "The number of partitions of 2n into exactly 2 parts. - _Colin Barker_, Mar 22 2015",
+ "The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- _Philippe A.J.G. Chevalier_, Dec 29 2015",
+ "Partial sums give A000217. - _Omar E. Pol_, Jul 26 2018",
+ "First differences are A000012 (the \"all 1's\" sequence). - _M. F. Hasler_, May 30 2020"
+ ],
+ "link": [
+ "N. J. A. Sloane, \u003ca href=\"/A001477/b001477.txt\"\u003eTable of n, a(n) for n = 0..500000\u003c/a\u003e",
+ "Paul Barry, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html\"\u003eA Catalan Transform and Related Transformations on Integer Sequences\u003c/a\u003e, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.",
+ "David Corneth, \u003ca href=\"https://www.youtube.com/watch?v=_rinkM0PCOQ\"\u003eCounting to 13999 visualized | showing changes per digit\u003c/a\u003e, YouTube video, 2019.",
+ "Hans Havermann, \u003ca href=\"/A001477/a001477.txt\"\u003eTable giving n and American English name for n, for 0 \u003c= n \u003c= 100999, without spaces or hyphens\u003c/a\u003e",
+ "Hans Havermann, \u003ca href=\"http://chesswanks.com/num/NumberNames.txt\"\u003eAmerican English number names to one million, without spaces or hyphens\u003c/a\u003e",
+ "Tanya Khovanova, \u003ca href=\"http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html\"\u003eRecursive Sequences\u003c/a\u003e",
+ "Luis Manuel Rivera, \u003ca href=\"http://arxiv.org/abs/1406.3081\"\u003eInteger sequences and k-commuting permutations\u003c/a\u003e, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.",
+ "László Németh, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html\"\u003eThe trinomial transform triangle\u003c/a\u003e, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also \u003ca href=\"https://arxiv.org/abs/1807.07109\"\u003earXiv:1807.07109\u003c/a\u003e [math.NT], 2018.",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/NaturalNumber.html\"\u003eNatural Number\u003c/a\u003e",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/NonnegativeInteger.html\"\u003eNonnegative Integer\u003c/a\u003e",
+ "\u003ca href=\"/index/Cor#core\"\u003eIndex entries for \"core\" sequences\u003c/a\u003e",
+ "\u003ca href=\"/index/Per#IntegerPermutation\"\u003eIndex entries for sequences that are permutations of the natural numbers\u003c/a\u003e",
+ "\u003ca href=\"/index/Rec#order_02\"\u003eIndex entries for linear recurrences with constant coefficients\u003c/a\u003e, signature (2,-1)."
+ ],
+ "formula": [
+ "a(n) = n.",
+ "a(0) = 0, a(n) = a(n-1) + 1.",
+ "G.f.: x/(1-x)^2.",
+ "Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001",
+ "When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - _Ralf Stephan_, Oct 17 2004",
+ "Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005",
+ "E.g.f.: x*e^x. - _Franklin T. Adams-Watters_, Sep 11 2005",
+ "a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - _Jaume Oliver Lafont_, May 07 2008",
+ "Alternating partial sums give A001057 = A000217 - 2*(A008794). - _Eric Desbiaux_, Oct 28 2008",
+ "a(n) = 2*A080425(n) + 3*A008611(n-3), n\u003e1. - _Eric Desbiaux_, Nov 15 2009",
+ "a(n) = Sum_{k\u003e=0} A030308(n,k)*2^k. - _Philippe Deléham_, Oct 20 2011",
+ "a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - _R. J. Mathar_, Jul 20 2012",
+ "a(n+1) = det(C(i+1,j), 1 \u003c= i, j \u003c= n), where C(n,k) are binomial coefficients. - _Mircea Merca_, Apr 06 2013",
+ "a(n-1) = floor(n/e^(1/n)) for n \u003e 0. - _Richard R. Forberg_, Jun 22 2013",
+ "a(n) = A000027(n) for all n\u003e0.",
+ "a(n) = floor(cot(1/(n+1))). - _Clark Kimberling_, Oct 08 2014",
+ "a(0)=0, a(n\u003e0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n\u003e0); a 1 to 1 correspondence between integers and naturals. - _Adriano Caroli_, Mar 29 2015"
+ ],
+ "example": [
+ "Triangular view:",
+ " 0",
+ " 1 2",
+ " 3 4 5",
+ " 6 7 8 9",
+ " 10 11 12 13 14",
+ " 15 16 17 18 19 20",
+ " 21 22 23 24 25 26 27",
+ " 28 29 30 31 32 33 34 35",
+ " 36 37 38 39 40 41 42 43 44",
+ " 45 46 47 48 49 50 51 52 53 54"
+ ],
+ "maple": [
+ "[ seq(n,n=0..100) ];"
+ ],
+ "mathematica": [
+ "Table[n, {n, 0, 100}] (* _Stefan Steinerberger_, Apr 08 2006 *)",
+ "LinearRecurrence[{2, -1}, {0, 1}, 77] (* _Robert G. Wilson v_, May 23 2013 *)",
+ "CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* _Robert G. Wilson v_, May 23 2013 *)"
+ ],
+ "program": [
+ "(MAGMA) [ n : n in [0..100]];",
+ "(PARI) A001477(n)=n /* first term is a(0) */",
+ "(Haskell)",
+ "a001477 = id",
+ "a001477_list = [0..] -- _Reinhard Zumkeller_, May 07 2012"
+ ],
+ "xref": [
+ "Cf. A000027 (n\u003e=1).",
+ "Cf. A000012 (first differences).",
+ "Partial sums of A057427. - _Jeremy Gardiner_, Sep 08 2002",
+ "Cf. A038608 (alternating signs), A001787 (binomial transform).",
+ "Cf. A055112.",
+ "Cf. Boustrophedon transforms: A231179, A000737.",
+ "Cf. A245422.",
+ "Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.",
+ "When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.",
+ "Cf. A000290."
+ ],
+ "keyword": "core,nonn,easy,mult,tabl",
+ "offset": "0,3",
+ "author": "_N. J. A. Sloane_",
+ "references": 727,
+ "revision": 273,
+ "time": "2021-02-15T22:43:26-05:00",
+ "created": "1991-04-30T03:00:00-04:00"
+ },
+ {
+ "number": 2260,
+ "data": "1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,2,3,4,5,6,1,2,3,4,5,6,7,1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,11,1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,1,2,3,4,5,6,7,8,9,10,11,12,13,14",
+ "name": "Triangle read by rows: T(n,k) = k for n \u003e= 1, k = 1..n.",
+ "comment": [
+ "Old name: integers 1 to k followed by integers 1 to k+1 etc. (a fractal sequence).",
+ "Start counting again and again.",
+ "This is a \"doubly fractal sequence\" - see the _Franklin T. Adams-Watters_ link.",
+ "The PARI functions t1, t2 can be used to read a square array T(n,k) (n \u003e= 1, k \u003e= 1) by antidiagonals downwards: n -\u003e T(t1(n), t2(n)). - _Michael Somos_, Aug 23 2002",
+ "Reading this sequence as the antidiagonals of a rectangular array, row n is (n,n,n,...); this is the weight array (Cf. A144112) of the array A127779 (rectangular). - _Clark Kimberling_, Sep 16 2008",
+ "The upper trim of an arbitrary fractal sequence s is s, but the lower trim of s, although a fractal sequence, need not be s itself. However, the lower trim of A002260 is A002260. (The upper trim of s is what remains after the first occurrence of each term is deleted; the lower trim of s is what remains after all 0's are deleted from the sequence s-1.) - _Clark Kimberling_, Nov 02 2009",
+ "Eigensequence of the triangle = A001710 starting (1, 3, 12, 60, 360,...). - _Gary W. Adamson_, Aug 02 2010",
+ "The triangle sums, see A180662 for their definitions, link this triangle of natural numbers with twenty-three different sequences, see the crossrefs. The mirror image of this triangle is A004736. - _Johannes W. Meijer_, Sep 22 2010",
+ "From _Paul Curtz_, Jul 25 2011: (Start)",
+ "Akiyama-Tanigawa algorithm from A000027(n) gives",
+ " 1, 2, 3, 4, 5, 6, 7, 8,",
+ " -1, -2, -3, -4, -5, -6, -7, -8,",
+ " 1, 2, 3, 4, 5, 6, 7, 8,",
+ " -1, -2, -3, -4, -5, -6, -7, -8.",
+ "By antidiagonals:",
+ " 1,",
+ " -1, 2,",
+ " 1, -2, 3,",
+ " -1, 2, -3, 4,",
+ " 1, -2, 3, -4, 5,",
+ " -1, 2, -3, 4, -5, 6.",
+ "Row sum = A016116. (End)",
+ "A002260 is the self-fission of the polynomial sequence (q(n,x)), where q(n,x) = x^n + x^(n-1) + ... + x + 1. See A193842 for the definition of fission. - _Clark Kimberling_, Aug 07 2011",
+ "Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A002260 is reluctant sequence of sequence 1,2,3,... (A000027). - _Boris Putievskiy_, Dec 12 2012",
+ "This is the maximal sequence of positive integers, such that once an integer k has occurred, the number of k's always exceeds the number of (k+1)'s for the remainder of the sequence, with the first occurrence of the integers being in order. - _Franklin T. Adams-Watters_, Oct 23 2013",
+ "A002260 are the k antidiagonal numerators of rationals in Cantor's proof of 1-to-1 correspondence between rationals and naturals; the denominators are k-numerator+1. - _Adriano Caroli_, Mar 24 2015",
+ "T(n,k) gives the distance to the largest triangular number \u003c n. - _Ctibor O. Zizka_, Apr 09 2020"
+ ],
+ "reference": [
+ "Clark Kimberling, \"Fractal sequences and interspersions,\" Ars Combinatoria 45 (1997) 157-168. (Introduces upper trimming, lower trimming, and signature sequences.)",
+ "M. Myers, Smarandache Crescendo Subsequences, R. H. Wilde, An Anthology in Memoriam, Bristol Banner Books, Bristol, 1998, p. 19.",
+ "F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006."
+ ],
+ "link": [
+ "N. J. A. Sloane, \u003ca href=\"/A002260/b002260.txt\"\u003eTable of n, a(n) for n = 1..11325\u003c/a\u003e",
+ "Franklin T. Adams-Watters, \u003ca href=\"/A002260/a002260.txt\"\u003eDoubly Fractal Sequences\u003c/a\u003e",
+ "Matin Amini and Majid Jahangiri, \u003ca href=\"https://arxiv.org/abs/1612.09481\"\u003eA Novel Proof for Kimberling’s Conjecture on Doubly Fractal Sequences\u003c/a\u003e, arXiv:1612.09481 [math.NT], 2017.",
+ "Bruno Berselli, \u003ca href=\"/A002260/a002260.jpg\"\u003eIllustration of the initial terms\u003c/a\u003e",
+ "Jerry Brown et al., \u003ca href=\"https://doi.org/10.1111/j.1949-8594.1997.tb17373.x\"\u003eProblem 4619\u003c/a\u003e, School Science and Mathematics (USA), Vol. 97(4), 1997, pp. 221-222.",
+ "Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, \u003ca href=\"http://docplayer.net/87034980-Vol-15-no-2-april-2017-dmmmsu-cas-science-monitor.html\"\u003eOn Fractal Sequences\u003c/a\u003e, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.",
+ "Clark Kimberling, \u003ca href=\"http://faculty.evansville.edu/ck6/integer/fractals.html\"\u003eFractal sequences\u003c/a\u003e",
+ "Clark Kimberling, \u003ca href=\"http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7321.pdf\"\u003eNumeration systems and fractal sequences\u003c/a\u003e, Acta Arithmetica 73 (1995) 103-117.",
+ "Boris Putievskiy, \u003ca href=\"http://arxiv.org/abs/1212.2732\"\u003eTransformations Integer Sequences And Pairing Functions\u003c/a\u003e arXiv:1212.2732 [math.CO], 2012.",
+ "F. Smarandache, \u003ca href=\"http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf\"\u003eSequences of Numbers Involved in Unsolved Problems\u003c/a\u003e.",
+ "Aaron Snook, \u003ca href=\"http://www.cs.cmu.edu/afs/cs/user/mjs/ftp/thesis-program/2012/theses/snook.pdf\"\u003eAugmented Integer Linear Recurrences\u003c/a\u003e, 2012. - _N. J. A. Sloane_, Dec 19 2012",
+ "Michael Somos, \u003ca href=\"/A073189/a073189.txt\"\u003eSequences used for indexing triangular or square arrays\u003c/a\u003e",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/SmarandacheSequences.html\"\u003eSmarandache Sequences.\u003c/a\u003e",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/UnitFraction.html\"\u003eUnit Fraction.\u003c/a\u003e"
+ ],
+ "formula": [
+ "a(n) = 1 + A002262(n).",
+ "n-th term is n - m*(m+1)/2 + 1, where m = floor((sqrt(8*n+1) - 1) / 2).",
+ "The above formula is for offset 0; for offset 1, use a(n) = n-m*(m+1)/2 where m = floor((-1+sqrt(8*n-7))/2). - _Clark Kimberling_, Jun 14 2011",
+ "a(k * (k + 1) / 2 + i) = i for k \u003e= 0 and 0 \u003c i \u003c= k + 1. - _Reinhard Zumkeller_, Aug 14 2001",
+ "a(n) = (2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2. - _Brian Tenneson_, Oct 11 2003",
+ "a(n) = n - binomial(floor((1+sqrt(8*n))/2), 2). - _Paul Barry_, May 25 2004",
+ "T(n,k) = A001511(A118413(n,k)); T(n,k) = A003602(A118416(n,k)). - _Reinhard Zumkeller_, Apr 27 2006",
+ "a(A000217(n)) = A000217(n) - A000217(n-1), a(A000217(n-1) + 1) = 1, a(A000217(n) - 1) = A000217(n) - A000217(n-1) - 1. - _Alexander R. Povolotsky_, May 28 2008",
+ "a(A169581(n)) = A038566(n). - _Reinhard Zumkeller_, Dec 02 2009",
+ "T(n,k) = Sum_{i=1..k} i*binomial(k,i)*binomial(n-k,n-i) (regarded as triangle, see the example). - _Mircea Merca_, Apr 11 2012",
+ "T(n,k) = Sum_{i=max(0,n+1-2*k)..n-k+1} (i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1). - _Vladimir Kruchinin_, Oct 18 2013",
+ "G.f.: x*y / ((1 - x) * (1 - x*y)^2) = Sum_{n,k\u003e0} T(n,k) * x^n * y^k. - _Michael Somos_, Sep 17 2014"
+ ],
+ "example": [
+ "First six rows:",
+ " 1",
+ " 1 2",
+ " 1 2 3",
+ " 1 2 3 4",
+ " 1 2 3 4 5",
+ " 1 2 3 4 5 6"
+ ],
+ "maple": [
+ "at:=0; for n from 1 to 150 do for i from 1 to n do at:=at+1; lprint(at,i); od: od: # _N. J. A. Sloane_, Nov 01 2006",
+ "seq(seq(i,i=1..k),k=1..13); # _Peter Luschny_, Jul 06 2009"
+ ],
+ "mathematica": [
+ "FoldList[{#1, #2} \u0026, 1, Range[2, 13]] // Flatten (* _Robert G. Wilson v_, May 10 2011 *)",
+ "Flatten[Table[Range[n],{n,20}]] (* _Harvey P. Dale_, Jun 20 2013 *)"
+ ],
+ "program": [
+ "(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* this sequence */",
+ "(Haskell)",
+ "a002260 n k = k",
+ "a002260_row n = [1..n]",
+ "a002260_tabl = iterate (\\row -\u003e map (+ 1) (0 : row)) [1]",
+ "-- _Reinhard Zumkeller_, Aug 04 2014, Jul 03 2012",
+ "(Maxima) T(n,k):=sum((i+k)*binomial(i+k-1,i)*binomial(k,n-i-k+1)*(-1)^(n-i-k+1),i,max(0,n+1-2*k),n-k+1); /* _Vladimir Kruchinin_, Oct 18 2013 */",
+ "(PARI) A002260(n)=n-binomial((sqrtint(8*n)+1)\\2,2) \\\\ _M. F. Hasler_, Mar 10 2014"
+ ],
+ "xref": [
+ "Cf. A000217, A001710, A002262, A003056, A004736 (ordinal transform), A025581, A056534, A094727, A127779.",
+ "Triangle sums (see the comments): A000217 (Row1, Kn11); A004526 (Row2); A000096 (Kn12); A055998 (Kn13); A055999 (Kn14); A056000 (Kn15); A056115 (Kn16); A056119 (Kn17); A056121 (Kn18); A056126 (Kn19); A051942 (Kn110); A101859 (Kn111); A132754 (Kn112); A132755 (Kn113); A132756 (Kn114); A132757 (Kn115); A132758 (Kn116); A002620 (Kn21); A000290 (Kn3); A001840 (Ca2); A000326 (Ca3); A001972 (Gi2); A000384 (Gi3).",
+ "Cf. A108872."
+ ],
+ "keyword": "nonn,easy,nice,tabl,look",
+ "offset": "1,3",
+ "author": "Angele Hamel (amh(AT)maths.soton.ac.uk)",
+ "ext": [
+ "More terms from _Reinhard Zumkeller_, Apr 27 2006",
+ "Incorrect program removed by _Franklin T. Adams-Watters_, Mar 19 2010",
+ "New name from _Omar E. Pol_, Jul 15 2012"
+ ],
+ "references": 417,
+ "revision": 192,
+ "time": "2021-02-05T12:04:16-05:00",
+ "created": "1996-12-11T03:00:00-05:00"
+ },
+ {
+ "number": 5349,
+ "id": "M0481",
+ "data": "1,2,3,4,5,6,7,8,9,10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,80,81,84,90,100,102,108,110,111,112,114,117,120,126,132,133,135,140,144,150,152,153,156,162,171,180,190,192,195,198,200,201,204",
+ "name": "Niven (or Harshad) numbers: numbers that are divisible by the sum of their digits.",
+ "comment": [
+ "z-Niven numbers are numbers n which are divisible by (A*s(n) + B) where A, B are integers and s(n) is sum of digits of n. Niven numbers have A = 1, B = 0. - _Ctibor O. Zizka_, Feb 23 2008",
+ "A070635(a(n)) = 0. A038186 is a subsequence. - _Reinhard Zumkeller_, Mar 10 2008",
+ "A049445 is a subsequence of this sequence. - _Ctibor O. Zizka_, Sep 06 2010",
+ "Complement of A065877; A188641(a(n)) = 1; A070635(a(n)) = 0. - _Reinhard Zumkeller_, Apr 07 2011",
+ "A001101, the Moran numbers, are a subsequence. - _Reinhard Zumkeller_, Jun 16 2011",
+ "A140866 gives the number of terms \u003c= 10^k. - _Robert G. Wilson v_, Oct 16 2012",
+ "The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - _Amiram Eldar_, Jul 10 2020"
+ ],
+ "reference": [
+ "Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.",
+ "R. E. Kennedy and C. N. Cooper, On the natural density of the Niven numbers, Abstract 816-11-219, Abstracts Amer. Math. Soc., 6 (1985), 17.",
+ "N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).",
+ "D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171."
+ ],
+ "link": [
+ "N. J. A. Sloane, \u003ca href=\"/A005349/b005349.txt\"\u003eTable of n, a(n) for n = 1..11872\u003c/a\u003e (all a(n) \u003c= 100000)",
+ "C. N. Cooper, R. E. Kennedy, \u003ca href=\"http://www.jstor.org/stable/2323194\"\u003eChebyshev's inequality and natural density\u003c/a\u003e, Amer. Math. Monthly 96 (1989), no. 2, 118-124.",
+ "Paul Dalenberg, Tom Edgar, \u003ca href=\"https://www.fq.math.ca/56-2.html\"\u003eConsecutive factorial base Niven numbers\u003c/a\u003e, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.",
+ "Jean-Marie De Koninck and Nicolas Doyon, \u003ca href=\"https://cs.uwaterloo.ca/journals/JIS/VOL6/Doyon/doyon.html\"\u003eLarge and Small Gaps Between Consecutive Niven Numbers\u003c/a\u003e, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.",
+ "R. K. Guy, \u003ca href=\"http://www.jstor.org/stable/2691503\"\u003eThe Second Strong Law of Small Numbers\u003c/a\u003e, Math. Mag, 63 (1990), no. 1, 3-20.",
+ "R. K. Guy, \u003ca href=\"/A005347/a005347.pdf\"\u003eThe Second Strong Law of Small Numbers\u003c/a\u003e, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]",
+ "R. E. Kennedy, \u003ca href=\"http://www.trottermath.net/numthry/nivennos.html\"\u003eNiven Numbers for Fun and Profit\u003c/a\u003e [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - _N. J. A. Sloane_, Mar 29 2018]",
+ "R. E. Kennedy and C. N. Cooper, \u003ca href=\"http://www.jstor.org/stable/2686395\"\u003eOn the natural density of the Niven numbers\u003c/a\u003e, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.",
+ "Project Euler, \u003ca href=\"https://projecteuler.net/problem=387\"\u003eHarshard Numbers: Problem 387\u003c/a\u003e",
+ "Gérard Villemin, \u003ca href=\"http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Harshad.htm\"\u003eNombres de Harshad\u003c/a\u003e (French)",
+ "Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, Eduard M. Albay, \u003ca href=\"http://www.dmmmsu-sluc.com/wp-content/uploads/2018/03/CAS-Monitor-2016-2017-1.pdf\"\u003eOn Harshad Number\u003c/a\u003e, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138.",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/Digit.html\"\u003eDigit\u003c/a\u003e and \u003ca href=\"http://mathworld.wolfram.com/HarshadNumber.html\"\u003eHarshad Numbers\u003c/a\u003e",
+ "Wikipedia, \u003ca href=\"http://en.wikipedia.org/wiki/Harshad_number\"\u003eHarshad number\u003c/a\u003e"
+ ],
+ "example": [
+ "195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5)."
+ ],
+ "maple": [
+ "s:=proc(n) local N:N:=convert(n,base,10):sum(N[j],j=1..nops(N)) end:p:=proc(n) if floor(n/s(n))=n/s(n) then n else fi end: seq(p(n),n=1..210); # _Emeric Deutsch_"
+ ],
+ "mathematica": [
+ "harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@ n] == 0; Select[ Range[1000], harshadQ] (* _Alonso del Arte_, Aug 04 2004 and modified by _Robert G. Wilson v_, Oct 16 2012 *)",
+ "Select[Range[300],Divisible[#,Total[IntegerDigits[#]]]\u0026] (* _Harvey P. Dale_, Sep 07 2015 *)"
+ ],
+ "program": [
+ "(Haskell)",
+ "a005349 n = a005349_list !! (n-1)",
+ "a005349_list = filter ((== 0) . a070635) [1..]",
+ "-- _Reinhard Zumkeller_, Aug 17 2011, Apr 07 2011",
+ "(MAGMA) [n: n in [1..250] | n mod \u0026+Intseq(n) eq 0]; // _Bruno Berselli_, May 28 2011",
+ "(MAGMA) [n: n in [1..250] | IsIntegral(n/\u0026+Intseq(n))]; // _Bruno Berselli_, Feb 09 2016",
+ "(PARI) is(n)=n%sumdigits(n)==0 \\\\ _Charles R Greathouse IV_, Oct 16 2012",
+ "(Python)",
+ "A005349 = [n for n in range(1,10**6) if not n % sum([int(d) for d in str(n)])] # _Chai Wah Wu_, Aug 22 2014",
+ "(Sage)",
+ "[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # _Freddy Barrera_, Jul 27 2018",
+ "(GAP) Filtered([1..230],n-\u003e n mod List(List([1..n],ListOfDigits),Sum)[n]=0); # _Muniru A Asiru_"
+ ],
+ "xref": [
+ "Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.",
+ "Cf. A001102 (a subsequence).",
+ "Cf. A118363 (for factorial-base analog)."
+ ],
+ "keyword": "nonn,base,nice,easy",
+ "offset": "1,2",
+ "author": "_N. J. A. Sloane_, _Robert G. Wilson v_",
+ "references": 267,
+ "revision": 145,
+ "time": "2020-07-10T03:50:09-04:00",
+ "created": "1991-07-11T03:00:00-04:00"
+ },
+ {
+ "number": 2262,
+ "data": "0,0,1,0,1,2,0,1,2,3,0,1,2,3,4,0,1,2,3,4,5,0,1,2,3,4,5,6,0,1,2,3,4,5,6,7,0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,10,0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,12,0,1,2,3,4,5,6,7,8,9,10,11,12,13",
+ "name": "Triangle read by rows: T(n,k), 0 \u003c= k \u003c= n, in which row n lists the first n+1 nonnegative integers.",
+ "comment": [
+ "The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. _N. J. A. Sloane_, Jul 17 2018",
+ "Old name: Integers 0 to n followed by integers 0 to n+1 etc.",
+ "a(n) = n - the largest triangular number \u003c= n. - _Amarnath Murthy_, Dec 25 2001",
+ "The PARI functions t1, t2 can be used to read a square array T(n,k) (n \u003e= 0, k \u003e= 0) by antidiagonals downwards: n -\u003e T(t1(n), t2(n)). - _Michael Somos_, Aug 23 2002",
+ "Values x of unique solution pair (x,y) to equation T(x+y) + x = n, where T(k)=A000217(k). - _Lekraj Beedassy_, Aug 21 2004",
+ "a(A000217(n)) = 0; a(A000096(n)) = n. - _Reinhard Zumkeller_, May 20 2009",
+ "Concatenation of the set representation of ordinal numbers, where the n-th ordinal number is represented by the set of all ordinals preceding n, 0 being represented by the empty set. - _Daniel Forgues_, Apr 27 2011",
+ "An integer sequence is nonnegative if and only if it is a subsequence of this sequence. - _Charles R Greathouse IV_, Sep 21 2011",
+ "a(A195678(n)) = A000040(n) and a(m) \u003c\u003e A000040(n) for m \u003c A195678(n), an example of the preceding comment. - _Reinhard Zumkeller_, Sep 23 2011",
+ "A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. A002262 is reluctant sequence of 0,1,2,3,... The nonnegative integers, A001477. - _Boris Putievskiy_, Dec 12 2012"
+ ],
+ "link": [
+ "Charles R Greathouse IV, \u003ca href=\"/A002262/b002262.txt\"\u003eRows n = 0..100, flattened\u003c/a\u003e",
+ "Boris Putievskiy, \u003ca href=\"http://arxiv.org/abs/1212.2732\"\u003eTransformations [Of] Integer Sequences And Pairing Functions\u003c/a\u003e, arXiv preprint arXiv:1212.2732 [math.CO], 2012.",
+ "Michael Somos, \u003ca href=\"/A073189/a073189.txt\"\u003eSequences used for indexing triangular or square arrays\u003c/a\u003e"
+ ],
+ "formula": [
+ "a(n) = A002260(n) - 1.",
+ "a(n) = n - (trinv(n)*(trinv(n)-1))/2; trinv := n -\u003e floor((1+sqrt(1+8*n))/2) (cf. A002024); # gives integral inverses of triangular numbers",
+ "a(n) = n - A000217(A003056(n)) = n - A057944(n). - _Lekraj Beedassy_, Aug 21 2004",
+ "a(n) = A140129(A023758(n+2)). - _Reinhard Zumkeller_, May 14 2008",
+ "a(n) = f(n,1) with f(n,m) = if n\u003cm then n else f(n-m,m+1). - _Reinhard Zumkeller_, May 20 2009",
+ "a(n) = (1/2)*(t - t^2 + 2*n), where t = floor(sqrt(2*n+1) + 1/2) = round(sqrt(2*n+1)). - _Ridouane Oudra_, Dec 01 2019"
+ ],
+ "example": [
+ "From _Daniel Forgues_, Apr 27 2011: (Start)",
+ "Examples of set-theoretic representation of ordinal numbers:",
+ " 0: {}",
+ " 1: {0} = {{}}",
+ " 2: {0, 1} = {0, {0}} = {{}, {{}}}",
+ " 3: {0, 1, 2} = {{}, {0}, {0, 1}} = ... = {{}, {{}}, {{}, {{}}}} (End)",
+ "From _Omar E. Pol_, Jul 15 2012: (Start)",
+ " 0;",
+ " 0, 1;",
+ " 0, 1, 2;",
+ " 0, 1, 2, 3;",
+ " 0, 1, 2, 3, 4;",
+ " 0, 1, 2, 3, 4, 5;",
+ " 0, 1, 2, 3, 4, 5, 6;",
+ " 0, 1, 2, 3, 4, 5, 6, 7;",
+ " 0, 1, 2, 3, 4, 5, 6, 7, 8;",
+ " 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;",
+ " 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;",
+ "(End)"
+ ],
+ "maple": [
+ "seq(seq(i,i=0..n),n=0..14); # _Peter Luschny_, Sep 22 2011",
+ "A002262 := n -\u003e n - binomial(floor((1/2)+sqrt(2*(1+n))),2);"
+ ],
+ "mathematica": [
+ "m[n_]:= Floor[(-1 + Sqrt[8n - 7])/2]",
+ "b[n_]:= n - m[n] (m[n] + 1)/2",
+ "Table[m[n], {n, 1, 105}] (* A003056 *)",
+ "Table[b[n], {n, 1, 105}] (* A002260 *)",
+ "Table[b[n] - 1, {n, 1, 120}] (* A002262 *)",
+ "(* _Clark Kimberling_, Jun 14 2011 *)",
+ "Flatten[Table[k, {n, 0, 14}, {k, 0, n}]] (* _Alonso del Arte_, Sep 21 2011 *)",
+ "Flatten[Table[Range[0,n], {n,0,15}]] (* _Harvey P. Dale_, Jan 31 2015 *)"
+ ],
+ "program": [
+ "(PARI) a(n)=n-binomial(round(sqrt(2+2*n)),2)",
+ "(PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262, this sequence */",
+ "(PARI) t2(n)=binomial(floor(3/2+sqrt(2+2*n)),2)-(n+1) /* A025581, cf. comment by Somos for reading arrays by antidiagonals */",
+ "(PARI) concat(vector(15,n,vector(n,i,i-1))) \\\\ _M. F. Hasler_, Sep 21 2011",
+ "(Haskell)",
+ "a002262 n k = a002262_tabl !! n !! k",
+ "a002262_row n = a002262_tabl !! n",
+ "a002262_tabl = map (enumFromTo 0) [0..]",
+ "a002262_list = concat a002262_tabl",
+ "-- _Reinhard Zumkeller_, Aug 05 2015, Jul 13 2012, Mar 07 2011",
+ "(Python)",
+ "for i in range(16):",
+ " for j in range(i):",
+ " print(j, end=\", \") # _Mohammad Saleh Dinparvar_, May 13 2020"
+ ],
+ "xref": [
+ "Cf. A002024, A002260, A004736, A025581, A025675, A025682.",
+ "Cf. A025691, A048645, A053186, A053645, A056558, A127324.",
+ "As a sequence, essentially same as A048151."
+ ],
+ "keyword": "nonn,tabl,easy,nice",
+ "offset": "0,6",
+ "author": "Angele Hamel (amh(AT)maths.soton.ac.uk)",
+ "ext": [
+ "New name from _Omar E. Pol_, Jul 15 2012",
+ "Typo in definition fixed by _Reinhard Zumkeller_, Aug 05 2015"
+ ],
+ "references": 225,
+ "revision": 131,
+ "time": "2020-05-27T10:26:43-04:00",
+ "created": "1996-12-11T03:00:00-05:00"
+ },
+ {
+ "number": 2473,
+ "id": "M0477 N0177",
+ "data": "1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,28,30,32,35,36,40,42,45,48,49,50,54,56,60,63,64,70,72,75,80,81,84,90,96,98,100,105,108,112,120,125,126,128,135,140,144,147,150,160,162,168,175,180,189,192",
+ "name": "7-smooth numbers: positive numbers whose prime divisors are all \u003c= 7.",
+ "comment": [
+ "Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.",
+ "Successive numbers k such that phi(210k) = 48k. - _Artur Jasinski_, Nov 05 2008",
+ "The divisors of 10! (A161466) are a finite subsequence. - _Reinhard Zumkeller_, Jun 10 2009",
+ "Numbers n such that A198487(n) \u003e 0 and A107698(n) \u003e 0. - _Jaroslav Krizek_, Nov 04 2011",
+ "A262401(a(n)) = a(n). - _Reinhard Zumkeller_, Sep 25 2015",
+ "Numbers which are products of single-digit numbers. - _N. J. A. Sloane_, Jul 02 2017",
+ "Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - _Richard Locke Peterson_, May 09 2020"
+ ],
+ "reference": [
+ "B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.",
+ "N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).",
+ "N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence)."
+ ],
+ "link": [
+ "Reinhard Zumkeller, \u003ca href=\"/A002473/b002473.txt\"\u003eTable of n, a(n) for n = 1..10000\u003c/a\u003e (first 5841 terms from N. J. A. Sloane)",
+ "Raphael Schumacher, \u003ca href=\"https://arxiv.org/abs/1608.06928\"\u003eThe Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers\u003c/a\u003e, arXiv preprint arXiv:1608.06928 [math.NT], 2016.",
+ "University of Ulm, \u003ca href=\"http://www.informatik.uni-ulm.de/acm/Locals/1996/number.sol\"\u003eThe first 5842 terms\u003c/a\u003e.",
+ "Eric Weisstein's World of Mathematics, \u003ca href=\"http://mathworld.wolfram.com/SmoothNumber.html\"\u003eSmooth Number\u003c/a\u003e."
+ ],
+ "formula": [
+ "A006530(a(n)) \u003c= 7. - _Reinhard Zumkeller_, Apr 01 2012",
+ "Sum_{n\u003e=1} 1/a(n) = Product_{primes p \u003c= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - _Amiram Eldar_, Sep 22 2020"
+ ],
+ "mathematica": [
+ "Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]\u003c=7\u0026]",
+ "aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* _Artur Jasinski_, Nov 05 2008 *)",
+ "mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # \u003c= 2^mxExp \u0026] (* _Harvey P. Dale_, Aug 13 2012 *)",
+ "mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}] (* _Robert G. Wilson v_, Aug 17 2012 *)"
+ ],
+ "program": [
+ "(PARI) test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1)",
+ "for(n=1,200,if(test(n),print1(n\",\")))",
+ "(PARI) is_A002473(n)=n\u003c11||vecmax(factor(n,7)[,1])\u003c8 \\\\ _M. F. Hasler_, Jan 16 2015",
+ "(PARI) list(lim)=my(v=List(),t); for(a=0,logint(lim\\1,7), for(b=0,logint(lim\\7^a,5), for(c=0,logint(lim\\7^a\\5^b,3), t=3^c*5^b*7^a; while(t\u003c=lim, listput(v,t); t\u003c\u003c=1)))); Set(v) \\\\ _Charles R Greathouse IV_, Feb 22 2017",
+ "(Haskell)",
+ "import Data.Set (singleton, deleteFindMin, fromList, union)",
+ "a002473 n = a002473_list !! (n-1)",
+ "a002473_list = f $ singleton 1 where",
+ " f s = x : f (s' `union` fromList (map (* x) [2,3,5,7]))",
+ " where (x, s') = deleteFindMin s",
+ "-- _Reinhard Zumkeller_, Mar 08 2014, Apr 02 2012, Apr 01 2012",
+ "(MAGMA) [n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // _Bruno Berselli_, Sep 24 2012"
+ ],
+ "xref": [
+ "Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.",
+ "Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.",
+ "Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.",
+ "Cf. A262401."
+ ],
+ "keyword": "nonn,easy,nice",
+ "offset": "1,2",
+ "author": "_N. J. A. Sloane_",
+ "ext": [
+ "More terms from _James A. Sellers_, Dec 23 1999",
+ "Additional comments from Michel Lecomte, Jun 09 2007",
+ "Edited by _M. F. Hasler_, Jan 16 2015"
+ ],
+ "references": 122,
+ "revision": 99,
+ "time": "2020-09-22T02:28:38-04:00",
+ "created": "1991-04-30T03:00:00-04:00"
+ },
+ {
+ "number": 72774,
+ "data": "1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,65,66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94,95,97",
+ "name": "Powers of squarefree numbers.",
+ "comment": [
+ "a(n) = A072775(n)^A072776(n); complement of A059404.",
+ "Essentially the same as A062770. - _R. J. Mathar_, Sep 25 2008",
+ "Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - _Reinhard Zumkeller_, Apr 06 2014",
+ "Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - _Gus Wiseman_, Apr 16 2018"
+ ],
+ "link": [
+ "Reinhard Zumkeller, \u003ca href=\"/A072774/b072774.txt\"\u003eTable of n, a(n) for n = 1..10000\u003c/a\u003e"
+ ],
+ "mathematica": [
+ "Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 \u0026] (* _Geoffrey Critzer_, Mar 30 2015 *)"
+ ],
+ "program": [
+ "(Haskell)",
+ "import Data.Map (empty, findMin, deleteMin, insert)",
+ "import qualified Data.Map.Lazy as Map (null)",
+ "a072774 n = a072774_list !! (n-1)",
+ "(a072774_list, a072775_list, a072776_list) = unzip3 $",
+ " (1, 1, 1) : f (tail a005117_list) empty where",
+ " f vs'@(v:vs) m",
+ " | Map.null m || xx \u003e v = (v, v, 1) :",
+ " f vs (insert (v^2) (v, 2) m)",
+ " | otherwise = (xx, bx, ex) :",
+ " f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)",
+ " where (xx, (bx, ex)) = findMin m",
+ "-- _Reinhard Zumkeller_, Apr 06 2014",
+ "(PARI) is(n)=ispower(n,,\u0026n); issquarefree(n) \\\\ _Charles R Greathouse IV_, Oct 16 2015"
+ ],
+ "xref": [
+ "Cf. A072777 (subsequence), A005117, A072778, A329332 (tabular arrangement).",
+ "A subsequence of A242414.",
+ "Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979."
+ ],
+ "keyword": "nonn",
+ "offset": "1,2",
+ "author": "_Reinhard Zumkeller_, Jul 10 2002",
+ "references": 89,
+ "revision": 33,
+ "time": "2020-01-09T10:43:18-05:00",
+ "created": "2003-05-16T03:00:00-04:00"
+ },
+ {
+ "number": 38566,
+ "data": "1,1,1,2,1,3,1,2,3,4,1,5,1,2,3,4,5,6,1,3,5,7,1,2,4,5,7,8,1,3,7,9,1,2,3,4,5,6,7,8,9,10,1,5,7,11,1,2,3,4,5,6,7,8,9,10,11,12,1,3,5,9,11,13,1,2,4,7,8,11,13,14,1,3,5,7,9,11,13,15,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16",
+ "name": "Numerators in canonical bijection from positive integers to positive rationals \u003c= 1: arrange fractions by increasing denominator then by increasing numerator.",
+ "comment": [
+ "Row n has length A000010(n).",
+ "Also numerators in canonical bijection from positive integers to all positive rational numbers: arrange fractions in triangle in which in the n-th row the phi(n) numbers are the fractions i/j with gcd(i,j) = 1, i+j=n, i=1..n-1, j=n-1..1. n\u003e=2. Denominators (A020653) are obtained by reversing each row.",
+ "Also triangle in which n-th row gives phi(n) numbers between 1 and n that are relatively prime to n.",
+ "a(n) = A002260(A169581(n)). - _Reinhard Zumkeller_, Dec 02 2009",
+ "A038610(n) = least common multiple of n-th row. - _Reinhard Zumkeller_, Sep 21 2013",
+ "Row n has sum A023896(n). - _Jamie Morken_, Dec 17 2019",
+ "This irregular triangle gives in row n the smallest positive reduced residue system modulo n, for n \u003e= 1. If one takes 0 for n = 1 it becomes the smallest nonnegative residue system modulo n. - _Wolfdieter Lang_, Feb 29 2020"
+ ],
+ "reference": [
+ "Richard Courant and Herbert Robbins. What Is Mathematics?, Oxford, 1941, pp. 79-80.",
+ "H. Lauwerier, Fractals, Princeton Univ. Press, p. 23."
+ ],
+ "link": [
+ "David Wasserman, \u003ca href=\"/A038566/b038566.txt\"\u003eTable of n, a(n) for n = 1..100001\u003c/a\u003e",
+ "Wolfdieter Lang, \u003ca href=\"/A038566/a038566.pdf\"\u003eRows of rationals, n=2..25.\u003c/a\u003e",
+ "\u003ca href=\"/index/Cor#core\"\u003eIndex entries for \"core\" sequences\u003c/a\u003e",
+ "\u003ca href=\"/index/Ra#rational\"\u003eIndex entries for sequences related to enumerating the rationals\u003c/a\u003e",
+ "\u003ca href=\"/index/St#Stern\"\u003eIndex entries for sequences related to Stern's sequences\u003c/a\u003e"
+ ],
+ "formula": [
+ "The n-th \"clump\" consists of the phi(n) integers \u003c= n and prime to n.",
+ "a(n+1) = A020652(n) for n \u003e 1. - _Georg Fischer_, Oct 27 2020"
+ ],
+ "example": [
+ "The beginning of the list of positive rationals \u003c= 1: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, .... This is A038566/A038567.",
+ "The beginning of the triangle giving all positive rationals: 1/1; 1/2, 2/1; 1/3, 3/1; 1/4, 2/3, 3/2, 4/1; 1/5, 5/1; 1/6, 2/5, 3/4, 4/3, 5/2, 6/1; .... This is A020652/A020653, with A020652(n) = A038566(n+1). [Corrected by _M. F. Hasler_, Mar 06 2020]",
+ "The beginning of the triangle in which n-th row gives numbers between 1 and n that are relatively prime to n:",
+ "n\\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18",
+ "1: 1",
+ "2: 1",
+ "3: 1 2",
+ "4: 1 3",
+ "5: 1 2 3 4",
+ "6: 1 5",
+ "7: 1 2 3 4 5 6",
+ "8: 1 3 5 7",
+ "9: 1 2 4 5 7 8",
+ "10: 1 3 7 9",
+ "11: 1 2 3 4 5 6 7 8 9 10",
+ "12: 1 5 7 11",
+ "13: 1 2 3 4 5 6 7 8 9 10 11 12",
+ "14: 1 3 5 9 11 13",
+ "15: 1 2 4 7 8 11 13 14",
+ "16: 1 3 5 7 9 11 13 15",
+ "17: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16",
+ "18: 1 5 7 11 13 17",
+ "19: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18",
+ "20: 1 3 7 9 11 13 17 19",
+ "... Reformatted. - _Wolfdieter Lang_, Jan 18 2017",
+ "------------------------------------------------------"
+ ],
+ "maple": [
+ "s := proc(n) local i,j,k,ans; i := 0; ans := [ ]; for j while i\u003cn do for k to j do if gcd(j,k) = 1 then ans := [ op(ans),k ]; i := i+1 fi od od; RETURN(ans); end; s(100);"
+ ],
+ "mathematica": [
+ "Flatten[Table[Flatten[Position[GCD[Table[Mod[j, w], {j, 1, w-1}], w], 1]], {w, 1, 100}], 2]"
+ ],
+ "program": [
+ "(Haskell)",
+ "a038566 n k = a038566_tabf !! (n-1) !! (k-1)",
+ "a038566_row n = a038566_tabf !! (n-1)",
+ "a038566_tabf=",
+ " zipWith (\\v ws -\u003e filter ((== 1) . (gcd v)) ws) [1..] a002260_tabl",
+ "a038566_list = concat a038566_tabf",
+ "-- _Reinhard Zumkeller_, Sep 21 2013, Feb 23 2012",
+ "(PARI) first(n)=my(v=List(),i,j);while(i\u003cn,for(k=1,j,if(gcd(j,k)==1, listput(v,k);i++));j++);Vec(v) \\\\ _Charles R Greathouse IV_, Feb 07 2013",
+ "(PARI) row(n) = select(x-\u003egcd(n, x)==1, [1..n]); \\\\ _Michel Marcus_, May 05 2020",
+ "(SageMath)",
+ "def aRow(n):",
+ " if n == 1: return 1",
+ " return [k for k in ZZ(n).coprime_integers(n+1)]",
+ "print(flatten([aRow(n) for n in range(1, 18)])) # _Peter Luschny_, Aug 17 2020"
+ ],
+ "xref": [
+ "Cf. A020652, A020653, A038566, A038567, A038568, A038569, A000010 (row lengths), A002088, A060837, A071970, A002260.",
+ "A054424 gives mapping to Stern-Brocot tree.",
+ "Row sums give rationals A111992(n)/A069220(n), n\u003e=1.",
+ "A112484 (primes, rows n \u003e=3)."
+ ],
+ "keyword": "nonn,frac,core,nice,tabf",
+ "offset": "1,4",
+ "author": "_N. J. A. Sloane_",
+ "ext": [
+ "More terms from _Erich Friedman_",
+ "Offset corrected by _Max Alekseyev_, Apr 26 2010"
+ ],
+ "references": 82,
+ "revision": 78,
+ "time": "2020-10-27T13:16:40-04:00",
+ "created": "1999-12-11T03:00:00-05:00"
+ },
+ {
+ "number": 28310,
+ "data": "1,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71",
+ "name": "Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x.",
+ "comment": [
+ "1 followed by the natural numbers.",
+ "Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).",
+ "Engel expansion of e (see A006784 for definition) [when offset by 1]. - _Henry Bottomley_, Dec 18 2000",
+ "Also the denominators of the series expansion of log(1+x). Numerators are A062157. - _Robert G. Wilson v_, Aug 14 2015",
+ "The right-shifted sequence (with a(0)=0) is an autosequence (of the first kind - see definition in links). - _Jean-François Alcover_, Mar 14 2017"
+ ],
+ "link": [
+ "Andrei Asinowski, Cyril Banderier, Valerie Roitner, \u003ca href=\"https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf\"\u003eGenerating functions for lattice paths with several forbidden patterns\u003c/a\u003e, (2019).",
+ "Olivia Nabawanda and Fanja Rakotondrajao, \u003ca href=\"https://arxiv.org/abs/2011.07304\"\u003eThe sets of flattened partitions with forbidden patterns\u003c/a\u003e, arXiv:2011.07304 [math.CO], 2020.",
+ "G. Nebe, E. M. Rains and N. J. A. Sloane, \u003ca href=\"http://neilsloane.com/doc/cliff2.html\"\u003eSelf-Dual Codes and Invariant Theory\u003c/a\u003e, Springer, Berlin, 2006.",
+ "Oeis Wiki, \u003ca href=\"https://oeis.org/wiki/Autosequence\"\u003eAutosequence\u003c/a\u003e",
+ "E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (\u003ca href=\"http://neilsloane.com/doc/self.txt\"\u003eAbstract\u003c/a\u003e, \u003ca href=\"http://neilsloane.com/doc/self.pdf\"\u003epdf\u003c/a\u003e, \u003ca href=\"http://neilsloane.com/doc/self.ps\"\u003eps\u003c/a\u003e).",
+ "Michael Somos, \u003ca href=\"http://cis.csuohio.edu/~somos/rfmc.txt\"\u003eRational Function Multiplicative Coefficients\u003c/a\u003e",
+ "\u003ca href=\"/index/Rec#order_02\"\u003eIndex entries for linear recurrences with constant coefficients\u003c/a\u003e, signature (2,-1).",
+ "\u003ca href=\"/index/Mo#Molien\"\u003eIndex entries for Molien series\u003c/a\u003e",
+ "\u003ca href=\"/index/El#Engel\"\u003eIndex entries for sequences related to Engel expansions\u003c/a\u003e"
+ ],
+ "formula": [
+ "Binomial transform is A005183. - _Paul Barry_, Jul 21 2003",
+ "G.f.: (1 - x + x^2) / (1 - x)^2 = (1 - x^6) /((1 - x) * (1 - x^2) * (1 - x^3)) = (1 + x^3) / ((1 - x) * (1 - x^2)). a(0) = 1, a(n) = n if n\u003e0.",
+ "Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - _Michael Somos_ Jul 30 2006",
+ "G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x)))). - _Michael Somos_, Apr 05 2012",
+ "G.f. of A112934(x) = 1 / (1 - a(0)*x / (1 - a(1)*x / ...)). - _Michael Somos_, Apr 05 2012",
+ "a(n) = A000027(n) unless n=0.",
+ "a(n) = Sum_{k, 0\u003c=k\u003c=n} A123110(n,k). - _Philippe Deléham_, Oct 06 2009",
+ "E.g.f: 1+x*exp(x). - _Wolfdieter Lang_, May 03 2010",
+ "a(n) = sqrt(floor[A204503(n+3)/9]). - _M. F. Hasler_, Jan 16 2012",
+ "E.g.f.: 1-x + x*E(0), where E(k) = 2 + x/(2*k+1 - x/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Dec 24 2013",
+ "a(n) = A001477(n) + A000007(n). - _Miko Labalan_, Dec 12 2015 (See the first comment.)"
+ ],
+ "example": [
+ "1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + 9*x^9 + ..."
+ ],
+ "maple": [
+ "a:= n-\u003e `if`(n=0, 1, n):",
+ "seq(a(n), n=0..60);"
+ ],
+ "mathematica": [
+ "f[0] = 0; f[1] = 1; f[2] = 1; f[3] = 1;",
+ "f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 3]];",
+ "Table[f[n], {n, 0, 50}] (* _Roger L. Bagula_, Feb 13 2010 *)",
+ "Denominator@ CoefficientList[ Series[ Log[1 + x], {x, 0, 75}], x] (* or *)",
+ "CoefficientList[ Series[(1 - x + x^2)/(1 - x)^2, {x, 0, 75}], x] (* _Robert G. Wilson v_, Aug 14 2015 *)"
+ ],
+ "program": [
+ "(PARI) {a(n) = (n==0) + max(n, 0)} /* _Michael Somos_, Feb 02 2004 */",
+ "(PARI) A028310(n)=n+!n \\\\ _M. F. Hasler_, Jan 16 2012",
+ "(Haskell)",
+ "a028310 n = 0 ^ n + n",
+ "a028310_list = 1 : [1..] -- _Reinhard Zumkeller_, Nov 06 2012"
+ ],
+ "xref": [
+ "Cf. A000027, A112934, A004001, A005229, A212393, A000660 (boustrophedon transform)."
+ ],
+ "keyword": "nonn,easy,mult",
+ "offset": "0,3",
+ "author": "_N. J. A. Sloane_",
+ "references": 79,
+ "revision": 117,
+ "time": "2021-03-12T22:24:41-05:00",
+ "created": "1999-12-11T03:00:00-05:00"
+ }
+ ]
+}
\ No newline at end of file
diff --git a/data/1,4,17,73,31.json b/data/1,4,17,73,31.json
new file mode 100644
index 0000000..946e311
--- /dev/null
+++ b/data/1,4,17,73,31.json
@@ -0,0 +1,7 @@
+{
+ "greeting": "Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/",
+ "query": "1,4,17,73,315,1362,5895,25528,110579,479068,2075683,8993897,",
+ "count": 0,
+ "start": 0,
+ "results": null
+}
\ No newline at end of file
diff --git a/data/BIGLIST.txt b/data/BIGLIST.txt
new file mode 100644
index 0000000..c558e15
--- /dev/null
+++ b/data/BIGLIST.txt
@@ -0,0 +1,749 @@
+=================
+BinomialTriangle
+
+Triangle: A007318 [1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
+Reverse: A007318 [1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
+Inverse: A007318 [1, -1, 1, 1, -2, 1, -1, 3, -3, 1]
+RevInv: A007318 [1, 1, -1, 1, -2, 1, 1, -3, 3, -1]
+InvRev: A007318 [1, -1, 1, 1, -2, 1, -1, 3, -3, 1]
+Diagonal: A011973 [1, 1, 1, 1, 1, 2, 1, 3, 1, 1]
+Sum: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+EvenSum: A011782 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+OddSum: A131577 [0, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: C000045 [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
+Central: A000984 [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+NegHalf: A000012 [1, -1, 1, -1, 1, -1, 1, -1, 1, -1]
+TransUnos: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
+TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
+TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]
+PolyVal2: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+PolyVal3: A000302 [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]
+
+=================
+CatalanTriangle
+
+Triangle: A053121 [1, 0, 1, 1, 0, 1, 0, 2, 0, 1]
+Reverse: A052173 [1, 1, 0, 1, 0, 1, 1, 0, 2, 0]
+Inverse: A049310 [1, 0, 1, -1, 0, 1, 0, -2, 0, 1]
+RevInv: A053119 [1, 1, 0, 1, 0, -1, 1, 0, -2, 0]
+Diagonal: nothing [1, 0, 1, 1, 0, 0, 2, 2, 1, 0]
+Sum: A001405 [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
+EvenSum: A126869 [1, 0, 2, 0, 6, 0, 20, 0, 70, 0]
+OddSum: A138364 [0, 1, 0, 3, 0, 10, 0, 35, 0, 126]
+AltSum: A001405 [1, -1, 2, -3, 6, -10, 20, -35, 70, -126]
+DiagSum: nothing [1, 0, 2, 0, 5, 0, 14, 0, 42, 0]
+Central: nothing [1, 0, 3, 0, 20, 0, 154, 0, 1260, 0]
+LeftSide: A126120 [1, 0, 1, 0, 2, 0, 5, 0, 14, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A121724 [1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289]
+NegHalf: A121724 [1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289]
+TransUnos: A001405 [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
+TransAlts: A001405 [1, -1, 2, -3, 6, -10, 20, -35, 70, -126]
+TransSqrs: nothing [0, 1, 4, 11, 28, 66, 152, 339, 748, 1622]
+TransNat0: A045621 [0, 1, 2, 5, 10, 22, 44, 93, 186, 386]
+TransNat1: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+PolyVal2: A054341 [1, 2, 5, 12, 30, 74, 185, 460, 1150, 2868]
+PolyVal3: A126931 [1, 3, 10, 33, 110, 366, 1220, 4065, 13550, 45162]
+
+=================
+EulerianTriangle
+
+Triangle: A173018 [1, 1, 0, 1, 1, 0, 1, 4, 1, 0]
+Reverse: A123125 [1, 0, 1, 0, 1, 1, 0, 1, 4, 1]
+InvRev: nothing [1, 0, 1, 0, -1, 1, 0, 3, -4, 1]
+Diagonal: nothing [1, 1, 1, 0, 1, 1, 1, 4, 0, 1]
+Sum: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+EvenSum: A128103 [1, 1, 1, 2, 12, 68, 360, 2384, 20160, 185408]
+OddSum: A262745 [0, 0, 1, 4, 12, 52, 360, 2656, 20160, 177472]
+AltSum: A009006 [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
+DiagSum: A000800 [1, 1, 1, 2, 5, 13, 38, 125, 449, 1742]
+Central: A180056 [1, 1, 11, 302, 15619, 1310354, 162512286, 27971176092, 6382798925475, 1865385657780650]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+PosHalf: A000670 [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
+NegHalf: A087674 [1, 1, -1, -3, 15, 21, -441, 477, 19935, -101979]
+TransUnos: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+TransAlts: A009006 [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
+TransSqrs: nothing [0, 0, 1, 8, 64, 540, 4920, 48720, 524160, 6108480]
+TransNat0: B180119 [0, 0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520]
+TransNat1: C001710 [1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400]
+PolyVal2: A000670 [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
+PolyVal3: A122704 [1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216]
+
+=================
+FibonacciTriangle
+
+Triangle: A193737 [1, 1, 1, 1, 2, 1, 2, 4, 3, 1]
+Reverse: A193736 [1, 1, 1, 1, 2, 1, 1, 3, 4, 2]
+Inverse: nothing [1, -1, 1, 1, -2, 1, -1, 2, -3, 1]
+RevInv: nothing [1, 1, -1, 1, -2, 1, 1, -3, 2, -1]
+Diagonal: B119473 [1, 1, 1, 1, 2, 2, 3, 4, 1, 5]
+Sum: A052542 [1, 2, 4, 10, 24, 58, 140, 338, 816, 1970]
+EvenSum: A215928 [1, 1, 2, 5, 12, 29, 70, 169, 408, 985]
+OddSum: A000129 [0, 1, 2, 5, 12, 29, 70, 169, 408, 985]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: A011782 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+Central: A330793 [1, 2, 8, 36, 170, 826, 4088, 20496, 103752, 529100]
+LeftSide: A324969 [1, 1, 1, 2, 3, 5, 8, 13, 21, 34]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A330795 [1, 3, 9, 39, 153, 615, 2457, 9831, 39321, 157287]
+NegHalf: B006131 [1, -1, 1, -5, 9, -29, 65, -181, 441, -1165]
+TransUnos: A052542 [1, 2, 4, 10, 24, 58, 140, 338, 816, 1970]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: nothing [0, 1, 6, 25, 92, 313, 1010, 3137, 9464, 27905]
+TransNat0: A119915 [0, 1, 4, 13, 40, 117, 332, 921, 2512, 6761]
+TransNat1: A331321 [1, 3, 8, 23, 64, 175, 472, 1259, 3328, 8731]
+PolyVal2: A052906 [1, 3, 9, 30, 99, 327, 1080, 3567, 11781, 38910]
+PolyVal3: nothing [1, 4, 16, 68, 288, 1220, 5168, 21892, 92736, 392836]
+
+=================
+LaguerreTriangle
+
+Triangle: A021009 [1, 1, 1, 2, 4, 1, 6, 18, 9, 1]
+Reverse: A021010 [1, 1, 1, 1, 4, 2, 1, 9, 18, 6]
+Inverse: A021009 [1, -1, 1, 2, -4, 1, -6, 18, -9, 1]
+RevInv: A021010 [1, 1, -1, 1, -4, 2, 1, -9, 18, -6]
+Diagonal: A084950 [1, 1, 2, 1, 6, 4, 24, 18, 1, 120]
+Sum: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+EvenSum: A331325 [1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257]
+OddSum: A331326 [0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857]
+AltSum: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+DiagSum: C001040 [1, 1, 3, 10, 43, 225, 1393, 9976, 81201, 740785]
+Central: A295383 [1, 4, 72, 2400, 117600, 7620480, 614718720, 59364264960, 6678479808000, 857813628672000]
+LeftSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025167 [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523]
+NegHalf: A025166 [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063, 84149983]
+TransUnos: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+TransAlts: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+TransSqrs: A105219 [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049]
+TransNat0: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat1: C000262 [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091]
+PolyVal2: A087912 [1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856]
+PolyVal3: A277382 [1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164]
+
+=================
+LahTriangle
+
+Triangle: A111596 [1, 0, 1, 0, 2, 1, 0, 6, 6, 1]
+Reverse: nothing [1, 1, 0, 1, 2, 0, 1, 6, 6, 0]
+Inverse: A111596 [1, 0, 1, 0, -2, 1, 0, 6, -6, 1]
+RevInv: nothing [1, 1, 0, 1, -2, 0, 1, -6, 6, 0]
+Diagonal: A330609 [1, 0, 0, 1, 0, 2, 0, 6, 1, 0]
+Sum: A000262 [1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]
+EvenSum: A088312 [1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536]
+OddSum: A088313 [0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017]
+AltSum: A111884 [1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]
+DiagSum: A001053 [1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660]
+Central: A187535 [1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480, 3339239904000, 428906814336000]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025168 [1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177, 417787921]
+NegHalf: A318223 [1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151]
+TransUnos: A000262 [1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]
+TransAlts: A111884 [1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]
+TransSqrs: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat0: A052852 [0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561]
+TransNat1: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+PolyVal2: A052897 [1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888, 42483968]
+PolyVal3: A255806 [1, 3, 15, 99, 801, 7623, 83079, 1017495, 13808097, 205374123]
+
+=================
+MotzkinTriangle
+
+Triangle: A064189 [1, 1, 1, 2, 2, 1, 4, 5, 3, 1]
+Reverse: A026300 [1, 1, 1, 1, 2, 2, 1, 3, 5, 4]
+Inverse: A101950 [1, -1, 1, 0, -2, 1, 1, 1, -3, 1]
+RevInv: nothing [1, 1, -1, 1, -2, 0, 1, -3, 1, 1]
+Diagonal: A106489 [1, 1, 2, 1, 4, 2, 9, 5, 1, 21]
+Sum: C005773 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046]
+EvenSum: A002426 [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139]
+OddSum: B005717 [0, 1, 2, 6, 16, 45, 126, 357, 1016, 2907]
+AltSum: A005043 [1, 0, 1, 1, 3, 6, 15, 36, 91, 232]
+DiagSum: nothing [1, 1, 3, 6, 15, 36, 91, 232, 603, 1585]
+Central: A026302 [1, 2, 9, 44, 230, 1242, 6853, 38376, 217242, 1239980]
+LeftSide: A001006 [1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A330799 [1, 3, 13, 59, 285, 1419, 7245, 37659, 198589, 1059371]
+NegHalf: A330800 [1, -1, 5, -17, 77, -345, 1653, -8097, 40733, -208553]
+TransUnos: C005773 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046]
+TransAlts: A005043 [1, 0, 1, 1, 3, 6, 15, 36, 91, 232]
+TransSqrs: nothing [0, 1, 6, 26, 100, 361, 1254, 4245, 14108, 46247]
+TransNat0: A330796 [0, 1, 4, 14, 46, 147, 462, 1437, 4438, 13637]
+TransNat1: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+PolyVal2: A059738 [1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629]
+PolyVal3: nothing [1, 4, 17, 73, 315, 1362, 5895, 25528, 110579, 479068]
+
+=================
+NarayanaTriangle
+
+Triangle: A090181 [1, 0, 1, 0, 1, 1, 0, 1, 3, 1]
+Reverse: A131198 [1, 1, 0, 1, 1, 0, 1, 3, 1, 0]
+Inverse: nothing [1, 0, 1, 0, -1, 1, 0, 2, -3, 1]
+RevInv: nothing [1, 1, 0, 1, -1, 0, 1, -3, 2, 0]
+Diagonal: nothing [1, 0, 0, 1, 0, 1, 0, 1, 1, 0]
+Sum: A000108 [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
+EvenSum: B071688 [1, 0, 1, 3, 7, 20, 66, 217, 715, 2424]
+OddSum: B071684 [0, 1, 1, 2, 7, 22, 66, 212, 715, 2438]
+AltSum: A090192 [1, -1, 0, 1, 0, -2, 0, 5, 0, -14]
+DiagSum: nothing [1, 0, 1, 1, 2, 4, 8, 17, 37, 82]
+Central: A125558 [1, 1, 6, 50, 490, 5292, 60984, 736164, 9202050, 118195220]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+NegHalf: A154825 [1, 1, -1, -1, 5, -3, -21, 51, 41, -391]
+TransUnos: A000108 [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
+TransAlts: A090192 [1, -1, 0, 1, 0, -2, 0, 5, 0, -14]
+TransSqrs: B141222 [0, 1, 5, 22, 95, 406, 1722, 7260, 30459, 127270]
+TransNat0: B001700 [0, 1, 3, 10, 35, 126, 462, 1716, 6435, 24310]
+TransNat1: A189176 [1, 2, 5, 15, 49, 168, 594, 2145, 7865, 29172]
+PolyVal2: A006318 [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]
+PolyVal3: A047891 [1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322]
+
+=================
+SchröderBTriangle
+
+Triangle: A122538 [1, 0, 1, 0, 2, 1, 0, 6, 4, 1]
+Reverse: nothing [1, 1, 0, 1, 2, 0, 1, 4, 6, 0]
+Inverse: A122542 [1, 0, 1, 0, -2, 1, 0, 2, -4, 1]
+RevInv: A266213 [1, 1, 0, 1, -2, 0, 1, -4, 2, 0]
+Diagonal: nothing [1, 0, 0, 1, 0, 2, 0, 6, 1, 0]
+Sum: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+EvenSum: nothing [1, 0, 1, 4, 17, 76, 353, 1688, 8257, 41128]
+OddSum: B010683 [0, 1, 2, 7, 28, 121, 550, 2591, 12536, 61921]
+AltSum: B001003 [1, -1, -1, -3, -11, -45, -197, -903, -4279, -20793]
+DiagSum: nothing [1, 0, 1, 2, 7, 26, 107, 468, 2141, 10124]
+Central: A103885 [1, 2, 16, 146, 1408, 14002, 142000, 1459810, 15158272, 158611106]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: B330802 [1, 1, 5, 33, 253, 2121, 18853, 174609, 1667021, 16290969]
+NegHalf: B330803 [1, 1, -3, 17, -123, 1001, -8739, 79969, -756939, 7349657]
+TransUnos: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+TransAlts: B001003 [1, -1, -1, -3, -11, -45, -197, -903, -4279, -20793]
+TransSqrs: A065096 [0, 1, 6, 31, 156, 785, 3978, 20335, 104856, 545073]
+TransNat0: B239204 [0, 1, 4, 17, 76, 353, 1688, 8257, 41128, 207905]
+TransNat1: A010683 [1, 2, 7, 28, 121, 550, 2591, 12536, 61921, 310954]
+PolyVal2: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+PolyVal3: nothing [1, 3, 15, 81, 453, 2583, 14907, 86733, 507561, 2982987]
+
+=================
+SchröderLTriangle
+
+Triangle: A172094 [1, 1, 1, 3, 4, 1, 11, 17, 7, 1]
+Reverse: nothing [1, 1, 1, 1, 4, 3, 1, 7, 17, 11]
+Inverse: A331969 [1, -1, 1, 1, -4, 1, -1, 11, -7, 1]
+RevInv: nothing [1, 1, -1, 1, -4, 1, 1, -7, 11, -1]
+Diagonal: nothing [1, 1, 3, 1, 11, 4, 45, 17, 1, 197]
+Sum: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+EvenSum: B225887 [1, 1, 4, 18, 86, 426, 2162, 11166, 58438, 309042]
+OddSum: B225887 [0, 1, 4, 18, 86, 426, 2162, 11166, 58438, 309042]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: nothing [1, 1, 4, 15, 63, 280, 1297, 6193, 30268, 150687]
+Central: nothing [1, 4, 40, 458, 5558, 69660, 891154, 11563214, 151605142, 2003523032]
+LeftSide: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A331328 [1, 3, 21, 171, 1509, 13995, 134277, 1320651, 13237221, 134682219]
+NegHalf: B330802 [1, -1, 5, -33, 253, -2121, 18853, -174609, 1667021, -16290969]
+TransUnos: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: nothing [0, 1, 8, 54, 342, 2098, 12634, 75190, 443934, 2606330]
+TransNat0: nothing [0, 1, 6, 34, 190, 1058, 5890, 32822, 183158, 1023658]
+TransNat1: nothing [1, 3, 14, 70, 362, 1910, 10214, 55154, 300034, 1641742]
+PolyVal2: nothing [1, 3, 15, 81, 453, 2583, 14907, 86733, 507561, 2982987]
+PolyVal3: nothing [1, 4, 24, 152, 984, 6440, 42408, 280312, 1857336, 12326792]
+
+=================
+StirlingCycleTriangle
+
+Triangle: A048994 [1, 0, 1, 0, 1, 1, 0, 2, 3, 1]
+Reverse: A054654 [1, 1, 0, 1, 1, 0, 1, 3, 2, 0]
+Inverse: A048993 [1, 0, 1, 0, -1, 1, 0, 1, -3, 1]
+RevInv: A106800 [1, 1, 0, 1, -1, 0, 1, -3, 1, 0]
+Diagonal: A331327 [1, 0, 0, 1, 0, 1, 0, 2, 1, 0]
+Sum: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+EvenSum: A105752 [1, 0, 1, 3, 12, 60, 360, 2520, 20160, 181440]
+OddSum: D001710 [0, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440]
+AltSum: A019590 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: nothing [1, 0, 1, 1, 3, 9, 36, 176, 1030, 7039]
+Central: A187646 [1, 1, 11, 225, 6769, 269325, 13339535, 790943153, 54631129553, 4308105301929]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A001147 [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
+NegHalf: A330797 [1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025]
+TransUnos: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+TransAlts: A019590 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: B151881 [0, 1, 5, 23, 120, 724, 5012, 39332, 345832, 3371976]
+TransNat0: A000254 [0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576]
+TransNat1: A000774 [1, 2, 5, 17, 74, 394, 2484, 18108, 149904, 1389456]
+PolyVal2: C000142 [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
+PolyVal3: nothing [1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400]
+
+=================
+StirlingSetTriangle
+
+Triangle: A048993 [1, 0, 1, 0, 1, 1, 0, 1, 3, 1]
+Reverse: A106800 [1, 1, 0, 1, 1, 0, 1, 3, 1, 0]
+Inverse: A048994 [1, 0, 1, 0, -1, 1, 0, 2, -3, 1]
+RevInv: A054654 [1, 1, 0, 1, -1, 0, 1, -3, 2, 0]
+Diagonal: nothing [1, 0, 0, 1, 0, 1, 0, 1, 1, 0]
+Sum: A000110 [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
+EvenSum: A024430 [1, 0, 1, 3, 8, 25, 97, 434, 2095, 10707]
+OddSum: A024429 [0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440]
+AltSum: A000587 [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
+DiagSum: A171367 [1, 0, 1, 1, 2, 4, 9, 22, 58, 164]
+Central: A007820 [1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A004211 [1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441]
+NegHalf: A009235 [1, 1, -1, -1, 9, -23, -25, 583, -3087, 4401]
+TransUnos: A000110 [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
+TransAlts: A000587 [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
+TransSqrs: A033452 [0, 1, 5, 22, 99, 471, 2386, 12867, 73681, 446620]
+TransNat0: B005493 [0, 1, 3, 10, 37, 151, 674, 3263, 17007, 94828]
+TransNat1: C000110 [1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
+PolyVal2: A001861 [1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182]
+PolyVal3: A027710 [1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643]
+
+=================
+T008279
+
+Triangle: A008279 [1, 1, 1, 1, 2, 2, 1, 3, 6, 6]
+Reverse: A094587 [1, 1, 1, 2, 2, 1, 6, 6, 3, 1]
+InvRev: A128229 [1, -1, 1, 0, -2, 1, 0, 0, -3, 1]
+Diagonal: nothing [1, 1, 1, 1, 1, 2, 1, 3, 2, 1]
+Sum: A000522 [1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410]
+EvenSum: A087208 [1, 1, 3, 7, 37, 141, 1111, 5923, 62217, 426457]
+OddSum: B002747 [0, 1, 2, 9, 28, 185, 846, 7777, 47384, 559953]
+AltSum: A000166 [1, 0, 1, -2, 9, -44, 265, -1854, 14833, -133496]
+DiagSum: A122852 [1, 1, 2, 3, 6, 11, 24, 51, 122, 291]
+Central: A001813 [1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+PosHalf: A010842 [1, 3, 10, 38, 168, 872, 5296, 37200, 297856, 2681216]
+NegHalf: A000023 [1, -1, 2, -2, 8, 8, 112, 656, 5504, 49024]
+TransUnos: A000522 [1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410]
+TransAlts: A000166 [1, 0, 1, -2, 9, -44, 265, -1854, 14833, -133496]
+TransSqrs: nothing [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056, 64116657]
+TransNat0: A093964 [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281]
+TransNat1: A001339 [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809, 8877691]
+PolyVal2: A010844 [1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969, 306323443]
+PolyVal3: A010845 [1, 4, 25, 226, 2713, 40696, 732529, 15383110, 369194641, 9968255308]
+Main.TriangleTraitCard
+
+WARNING: replacing module TriangleTraitCard.
+
+=================
+BinomialTriangle
+
+Triangle: A007318 [1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
+Reverse: A007318 [1, 1, 1, 1, 2, 1, 1, 3, 3, 1]
+Inverse: A007318 [1, -1, 1, 1, -2, 1, -1, 3, -3, 1]
+RevInv: A007318 [1, 1, -1, 1, -2, 1, 1, -3, 3, -1]
+InvRev: A007318 [1, -1, 1, 1, -2, 1, -1, 3, -3, 1]
+Diagonal: A011973 [1, 1, 1, 1, 1, 2, 1, 3, 1, 1]
+Sum: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+EvenSum: A011782 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+OddSum: A131577 [0, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: C000045 [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
+Central: A000984 [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+NegHalf: A000012 [1, -1, 1, -1, 1, -1, 1, -1, 1, -1]
+TransUnos: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
+TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
+TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]
+PolyVal2: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+PolyVal3: A000302 [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]
+
+=================
+CatalanTriangle
+
+Triangle: A053121 [1, 0, 1, 1, 0, 1, 0, 2, 0, 1]
+Reverse: A052173 [1, 1, 0, 1, 0, 1, 1, 0, 2, 0]
+Inverse: A049310 [1, 0, 1, -1, 0, 1, 0, -2, 0, 1]
+RevInv: A053119 [1, 1, 0, 1, 0, -1, 1, 0, -2, 0]
+Diagonal: [ Info: Dowloaded 1,0,1,1,0,0,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 1, 0, 0, 2, 2, 1, 0]
+Sum: A001405 [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
+EvenSum: A126869 [1, 0, 2, 0, 6, 0, 20, 0, 70, 0]
+OddSum: A138364 [0, 1, 0, 3, 0, 10, 0, 35, 0, 126]
+AltSum: A001405 [1, -1, 2, -3, 6, -10, 20, -35, 70, -126]
+DiagSum: [ Info: Dowloaded 1,0,2,0,5,0,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 0, 2, 0, 5, 0, 14, 0, 42, 0]
+Central: [ Info: Dowloaded 1,0,3,0,20,0.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 3, 0, 20, 0, 154, 0, 1260, 0]
+LeftSide: A126120 [1, 0, 1, 0, 2, 0, 5, 0, 14, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A121724 [1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289]
+NegHalf: A121724 [1, 1, 5, 9, 45, 97, 485, 1145, 5725, 14289]
+TransUnos: A001405 [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]
+TransAlts: A001405 [1, -1, 2, -3, 6, -10, 20, -35, 70, -126]
+TransSqrs: [ Info: Dowloaded 0,1,4,11,28,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 4, 11, 28, 66, 152, 339, 748, 1622]
+TransNat0: A045621 [0, 1, 2, 5, 10, 22, 44, 93, 186, 386]
+TransNat1: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+PolyVal2: A054341 [1, 2, 5, 12, 30, 74, 185, 460, 1150, 2868]
+PolyVal3: A126931 [1, 3, 10, 33, 110, 366, 1220, 4065, 13550, 45162]
+
+=================
+EulerianTriangle
+
+Triangle: A173018 [1, 1, 0, 1, 1, 0, 1, 4, 1, 0]
+Reverse: A123125 [1, 0, 1, 0, 1, 1, 0, 1, 4, 1]
+InvRev: [ Info: Dowloaded 1,0,1,0,1,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 0, -1, 1, 0, 3, -4, 1]
+Diagonal: [ Info: Dowloaded 1,1,1,0,1,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, 1, 0, 1, 1, 1, 4, 0, 1]
+Sum: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+EvenSum: A128103 [1, 1, 1, 2, 12, 68, 360, 2384, 20160, 185408]
+OddSum: A262745 [0, 0, 1, 4, 12, 52, 360, 2656, 20160, 177472]
+AltSum: A009006 [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
+DiagSum: A000800 [1, 1, 1, 2, 5, 13, 38, 125, 449, 1742]
+Central: A180056 [1, 1, 11, 302, 15619, 1310354, 162512286, 27971176092, 6382798925475, 1865385657780650]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+PosHalf: A000670 [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
+NegHalf: A087674 [1, 1, -1, -3, 15, 21, -441, 477, 19935, -101979]
+TransUnos: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+TransAlts: A009006 [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
+TransSqrs: [ Info: Dowloaded 0,0,1,8,64,5.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 0, 1, 8, 64, 540, 4920, 48720, 524160, 6108480]
+TransNat0: B180119 [0, 0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520]
+TransNat1: C001710 [1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400]
+PolyVal2: A000670 [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]
+PolyVal3: A122704 [1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216]
+
+=================
+FibonacciTriangle
+
+Triangle: A193737 [1, 1, 1, 1, 2, 1, 2, 4, 3, 1]
+Reverse: A193736 [1, 1, 1, 1, 2, 1, 1, 3, 4, 2]
+Inverse: [ Info: Dowloaded 1,1,1,1,2,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, -1, 1, 1, -2, 1, -1, 2, -3, 1]
+RevInv: [ Info: Dowloaded 1,1,1,1,2,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 1, -1, 1, -2, 1, 1, -3, 2, -1, 1, -4, 4, 0, 1, 1, -5, 7]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 1, -1, 1, -2, 1, 1, -3, 2, -1]
+Diagonal: B119473 [1, 1, 1, 1, 2, 2, 3, 4, 1, 5]
+Sum: A052542 [1, 2, 4, 10, 24, 58, 140, 338, 816, 1970]
+EvenSum: A215928 [1, 1, 2, 5, 12, 29, 70, 169, 408, 985]
+OddSum: A000129 [0, 1, 2, 5, 12, 29, 70, 169, 408, 985]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: A011782 [1, 1, 2, 4, 8, 16, 32, 64, 128, 256]
+Central: A330793 [1, 2, 8, 36, 170, 826, 4088, 20496, 103752, 529100]
+LeftSide: A324969 [1, 1, 1, 2, 3, 5, 8, 13, 21, 34]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A330795 [1, 3, 9, 39, 153, 615, 2457, 9831, 39321, 157287]
+NegHalf: B006131 [1, -1, 1, -5, 9, -29, 65, -181, 441, -1165]
+TransUnos: A052542 [1, 2, 4, 10, 24, 58, 140, 338, 816, 1970]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: [ Info: Dowloaded 0,1,6,25,92,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 6, 25, 92, 313, 1010, 3137, 9464, 27905]
+TransNat0: A119915 [0, 1, 4, 13, 40, 117, 332, 921, 2512, 6761]
+TransNat1: A331321 [1, 3, 8, 23, 64, 175, 472, 1259, 3328, 8731]
+PolyVal2: A052906 [1, 3, 9, 30, 99, 327, 1080, 3567, 11781, 38910]
+PolyVal3: [ Info: Dowloaded 1,4,16,68,28.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 4, 16, 68, 288, 1220, 5168, 21892, 92736, 392836]
+
+=================
+LaguerreTriangle
+
+Triangle: A021009 [1, 1, 1, 2, 4, 1, 6, 18, 9, 1]
+Reverse: A021010 [1, 1, 1, 1, 4, 2, 1, 9, 18, 6]
+Inverse: A021009 [1, -1, 1, 2, -4, 1, -6, 18, -9, 1]
+RevInv: A021010 [1, 1, -1, 1, -4, 2, 1, -9, 18, -6]
+Diagonal: A084950 [1, 1, 2, 1, 6, 4, 24, 18, 1, 120]
+Sum: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+EvenSum: A331325 [1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257]
+OddSum: A331326 [0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857]
+AltSum: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+DiagSum: C001040 [1, 1, 3, 10, 43, 225, 1393, 9976, 81201, 740785]
+Central: A295383 [1, 4, 72, 2400, 117600, 7620480, 614718720, 59364264960, 6678479808000, 857813628672000]
+LeftSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025167 [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089, 2250495523]
+NegHalf: A025166 [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063, 84149983]
+TransUnos: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+TransAlts: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+TransSqrs: A105219 [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049]
+TransNat0: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat1: C000262 [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091]
+PolyVal2: A087912 [1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856]
+PolyVal3: A277382 [1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449, 490102164]
+
+=================
+LahTriangle
+
+Triangle: A111596 [1, 0, 1, 0, 2, 1, 0, 6, 6, 1]
+Reverse: nothing [1, 1, 0, 1, 2, 0, 1, 6, 6, 0]
+Inverse: A111596 [1, 0, 1, 0, -2, 1, 0, 6, -6, 1]
+RevInv: nothing [1, 1, 0, 1, -2, 0, 1, -6, 6, 0]
+Diagonal: A330609 [1, 0, 0, 1, 0, 2, 0, 6, 1, 0]
+Sum: A000262 [1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]
+EvenSum: A088312 [1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536]
+OddSum: A088313 [0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017]
+AltSum: A111884 [1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]
+DiagSum: A001053 [1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660]
+Central: A187535 [1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480, 3339239904000, 428906814336000]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025168 [1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177, 417787921]
+NegHalf: A318223 [1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151]
+TransUnos: A000262 [1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]
+TransAlts: A111884 [1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]
+TransSqrs: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat0: A052852 [0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561]
+TransNat1: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+PolyVal2: A052897 [1, 2, 8, 44, 304, 2512, 24064, 261536, 3173888, 42483968]
+PolyVal3: A255806 [1, 3, 15, 99, 801, 7623, 83079, 1017495, 13808097, 205374123]
+
+=================
+MotzkinTriangle
+
+Triangle: A064189 [1, 1, 1, 2, 2, 1, 4, 5, 3, 1]
+Reverse: A026300 [1, 1, 1, 1, 2, 2, 1, 3, 5, 4]
+Inverse: A101950 [1, -1, 1, 0, -2, 1, 1, 1, -3, 1]
+RevInv: nothing [1, 1, -1, 1, -2, 0, 1, -3, 1, 1]
+Diagonal: A106489 [1, 1, 2, 1, 4, 2, 9, 5, 1, 21]
+Sum: C005773 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046]
+EvenSum: A002426 [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139]
+OddSum: B005717 [0, 1, 2, 6, 16, 45, 126, 357, 1016, 2907]
+AltSum: A005043 [1, 0, 1, 1, 3, 6, 15, 36, 91, 232]
+DiagSum: [1, 1, 3, 6, 15, 36, 91, 232, 603, 1585, 4213, 11298, 30537, 83097, 227475, 625992, 1730787, 4805595]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 1, 3, 6, 15, 36, 91, 232, 603, 1585]
+Central: A026302 [1, 2, 9, 44, 230, 1242, 6853, 38376, 217242, 1239980]
+LeftSide: A001006 [1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A330799 [1, 3, 13, 59, 285, 1419, 7245, 37659, 198589, 1059371]
+NegHalf: A330800 [1, -1, 5, -17, 77, -345, 1653, -8097, 40733, -208553]
+TransUnos: C005773 [1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046]
+TransAlts: A005043 [1, 0, 1, 1, 3, 6, 15, 36, 91, 232]
+TransSqrs: [ Info: Dowloaded 0,1,6,26,100.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 6, 26, 100, 361, 1254, 4245, 14108, 46247]
+TransNat0: A330796 [0, 1, 4, 14, 46, 147, 462, 1437, 4438, 13637]
+TransNat1: A000244 [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]
+PolyVal2: A059738 [1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629]
+PolyVal3: [ Info: Dowloaded 1,4,17,73,31.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 4, 17, 73, 315, 1362, 5895, 25528, 110579, 479068]
+
+=================
+NarayanaTriangle
+
+Triangle: A090181 [1, 0, 1, 0, 1, 1, 0, 1, 3, 1]
+Reverse: A131198 [1, 1, 0, 1, 1, 0, 1, 3, 1, 0]
+Inverse: [ Info: Dowloaded 1,0,1,0,1,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -7, 12, -6, 1, 0, 39, -70]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 0, 1, 0, -1, 1, 0, 2, -3, 1]
+RevInv: [ Info: Dowloaded 1,1,0,1,1,0,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 12, -7, 0, 1, -10, 40]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 1, 0, 1, -1, 0, 1, -3, 2, 0]
+Diagonal: [ Info: Dowloaded 1,0,0,1,0,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 0, 1, 0, 1, 0, 1, 1, 0]
+Sum: A000108 [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
+EvenSum: B071688 [1, 0, 1, 3, 7, 20, 66, 217, 715, 2424]
+OddSum: B071684 [0, 1, 1, 2, 7, 22, 66, 212, 715, 2438]
+AltSum: A090192 [1, -1, 0, 1, 0, -2, 0, 5, 0, -14]
+DiagSum: [ Info: Dowloaded 1,0,1,1,2,4,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 1, 2, 4, 8, 17, 37, 82]
+Central: A125558 [1, 1, 6, 50, 490, 5292, 60984, 736164, 9202050, 118195220]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+NegHalf: A154825 [1, 1, -1, -1, 5, -3, -21, 51, 41, -391]
+TransUnos: A000108 [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]
+TransAlts: A090192 [1, -1, 0, 1, 0, -2, 0, 5, 0, -14]
+TransSqrs: B141222 [0, 1, 5, 22, 95, 406, 1722, 7260, 30459, 127270]
+TransNat0: B001700 [0, 1, 3, 10, 35, 126, 462, 1716, 6435, 24310]
+TransNat1: A189176 [1, 2, 5, 15, 49, 168, 594, 2145, 7865, 29172]
+PolyVal2: A006318 [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]
+PolyVal3: A047891 [1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322]
+
+=================
+SchröderBTriangle
+
+Triangle: A122538 [1, 0, 1, 0, 2, 1, 0, 6, 4, 1]
+Reverse: [ Info: Dowloaded 1,1,0,1,2,0,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, 8, 30]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 1, 0, 1, 2, 0, 1, 4, 6, 0]
+Inverse: A122542 [1, 0, 1, 0, -2, 1, 0, 2, -4, 1]
+RevInv: A266213 [1, 1, 0, 1, -2, 0, 1, -4, 2, 0]
+Diagonal: [ Info: Dowloaded 1,0,0,1,0,2,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 0, 1, 0, 2, 0, 6, 1, 0]
+Sum: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+EvenSum: [ Info: Dowloaded 1,0,1,4,17,7.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 4, 17, 76, 353, 1688, 8257, 41128]
+OddSum: B010683 [0, 1, 2, 7, 28, 121, 550, 2591, 12536, 61921]
+AltSum: B001003 [1, -1, -1, -3, -11, -45, -197, -903, -4279, -20793]
+DiagSum: [ Info: Dowloaded 1,0,1,2,7,26.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 2, 7, 26, 107, 468, 2141, 10124]
+Central: A103885 [1, 2, 16, 146, 1408, 14002, 142000, 1459810, 15158272, 158611106]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: B330802 [1, 1, 5, 33, 253, 2121, 18853, 174609, 1667021, 16290969]
+NegHalf: B330803 [1, 1, -3, 17, -123, 1001, -8739, 79969, -756939, 7349657]
+TransUnos: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+TransAlts: B001003 [1, -1, -1, -3, -11, -45, -197, -903, -4279, -20793]
+TransSqrs: A065096 [0, 1, 6, 31, 156, 785, 3978, 20335, 104856, 545073]
+TransNat0: B239204 [0, 1, 4, 17, 76, 353, 1688, 8257, 41128, 207905]
+TransNat1: A010683 [1, 2, 7, 28, 121, 550, 2591, 12536, 61921, 310954]
+PolyVal2: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+PolyVal3: [ Info: Dowloaded 1,3,15,81,45.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 3, 15, 81, 453, 2583, 14907, 86733, 507561, 2982987]
+
+=================
+SchröderLTriangle
+
+Triangle: A172094 [1, 1, 1, 3, 4, 1, 11, 17, 7, 1]
+Reverse: [ Info: Dowloaded 1,1,1,1,4,3,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, 1, 1, 4, 3, 1, 7, 17, 11]
+Inverse: A331969 [1, -1, 1, 1, -4, 1, -1, 11, -7, 1]
+RevInv: [ Info: Dowloaded 1,1,1,1,4,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, -1, 1, -4, 1, 1, -7, 11, -1]
+Diagonal: [ Info: Dowloaded 1,1,3,1,11,4.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, 3, 1, 11, 4, 45, 17, 1, 197]
+Sum: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+EvenSum: B225887 [1, 1, 4, 18, 86, 426, 2162, 11166, 58438, 309042]
+OddSum: B225887 [0, 1, 4, 18, 86, 426, 2162, 11166, 58438, 309042]
+AltSum: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: [ Info: Dowloaded 1,1,4,15,63,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, 4, 15, 63, 280, 1297, 6193, 30268, 150687]
+Central: [ Info: Dowloaded 1,4,40,458,5.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 4, 40, 458, 5558, 69660, 891154, 11563214, 151605142, 2003523032]
+LeftSide: A001003 [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A331328 [1, 3, 21, 171, 1509, 13995, 134277, 1320651, 13237221, 134682219]
+NegHalf: B330802 [1, -1, 5, -33, 253, -2121, 18853, -174609, 1667021, -16290969]
+TransUnos: A109980 [1, 2, 8, 36, 172, 852, 4324, 22332, 116876, 618084]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: [ Info: Dowloaded 0,1,8,54,342.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 8, 54, 342, 2098, 12634, 75190, 443934, 2606330]
+TransNat0: [ Info: Dowloaded 0,1,6,34,190.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 6, 34, 190, 1058, 5890, 32822, 183158, 1023658]
+TransNat1: [ Info: Dowloaded 1,3,14,70,36.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 3, 14, 70, 362, 1910, 10214, 55154, 300034, 1641742]
+PolyVal2: [ Info: Dowloaded 1,3,15,81,45.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 3, 15, 81, 453, 2583, 14907, 86733, 507561, 2982987]
+PolyVal3: [ Info: Dowloaded 1,4,24,152,9.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 4, 24, 152, 984, 6440, 42408, 280312, 1857336, 12326792]
+
+=================
+StirlingCycleTriangle
+
+Triangle: A048994 [1, 0, 1, 0, 1, 1, 0, 2, 3, 1]
+Reverse: A054654 [1, 1, 0, 1, 1, 0, 1, 3, 2, 0]
+Inverse: A048993 [1, 0, 1, 0, -1, 1, 0, 1, -3, 1]
+RevInv: A106800 [1, 1, 0, 1, -1, 0, 1, -3, 1, 0]
+Diagonal: A331327 [1, 0, 0, 1, 0, 1, 0, 2, 1, 0]
+Sum: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+EvenSum: A105752 [1, 0, 1, 3, 12, 60, 360, 2520, 20160, 181440]
+OddSum: D001710 [0, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440]
+AltSum: A019590 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0]
+DiagSum: [ Info: Dowloaded 1,0,1,1,3,9,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 1, 1, 3, 9, 36, 176, 1030, 7039]
+Central: A187646 [1, 1, 11, 225, 6769, 269325, 13339535, 790943153, 54631129553, 4308105301929]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A001147 [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
+NegHalf: A330797 [1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025]
+TransUnos: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+TransAlts: A019590 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: B151881 [0, 1, 5, 23, 120, 724, 5012, 39332, 345832, 3371976]
+TransNat0: A000254 [0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576]
+TransNat1: A000774 [1, 2, 5, 17, 74, 394, 2484, 18108, 149904, 1389456]
+PolyVal2: C000142 [1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
+PolyVal3: [ Info: Dowloaded 1,3,12,60,36.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+
+[1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000]
+┌ Warning: Not found in the local base but possibly in the OEIS!
+└ @ TrianglesUtils c:\Users\User\GitHub2021\IntegerTriangles.jl\src\TrianglesUtils.jl:142
+
+nothing [1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400]
+
+=================
+StirlingSetTriangle
+
+Triangle: A048993 [1, 0, 1, 0, 1, 1, 0, 1, 3, 1]
+Reverse: A106800 [1, 1, 0, 1, 1, 0, 1, 3, 1, 0]
+Inverse: A048994 [1, 0, 1, 0, -1, 1, 0, 2, -3, 1]
+RevInv: A054654 [1, 1, 0, 1, -1, 0, 1, -3, 2, 0]
+Diagonal: [ Info: Dowloaded 1,0,0,1,0,1,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 0, 0, 1, 0, 1, 0, 1, 1, 0]
+Sum: A000110 [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
+EvenSum: A024430 [1, 0, 1, 3, 8, 25, 97, 434, 2095, 10707]
+OddSum: A024429 [0, 1, 1, 2, 7, 27, 106, 443, 2045, 10440]
+AltSum: A000587 [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
+DiagSum: A171367 [1, 0, 1, 1, 2, 4, 9, 22, 58, 164]
+Central: A007820 [1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755]
+LeftSide: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A004211 [1, 1, 3, 11, 49, 257, 1539, 10299, 75905, 609441]
+NegHalf: A009235 [1, 1, -1, -1, 9, -23, -25, 583, -3087, 4401]
+TransUnos: A000110 [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]
+TransAlts: A000587 [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]
+TransSqrs: A033452 [0, 1, 5, 22, 99, 471, 2386, 12867, 73681, 446620]
+TransNat0: B005493 [0, 1, 3, 10, 37, 151, 674, 3263, 17007, 94828]
+TransNat1: C000110 [1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
+PolyVal2: A001861 [1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182]
+PolyVal3: A027710 [1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643]
+
+=================
+T008279
+
+Triangle: A008279 [1, 1, 1, 1, 2, 2, 1, 3, 6, 6]
+Reverse: A094587 [1, 1, 1, 2, 2, 1, 6, 6, 3, 1]
+InvRev: A128229 [1, -1, 1, 0, -2, 1, 0, 0, -3, 1]
+Diagonal: [ Info: Dowloaded 1,1,1,1,1,2,.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [1, 1, 1, 1, 1, 2, 1, 3, 2, 1]
+Sum: A000522 [1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410]
+EvenSum: A087208 [1, 1, 3, 7, 37, 141, 1111, 5923, 62217, 426457]
+OddSum: B002747 [0, 1, 2, 9, 28, 185, 846, 7777, 47384, 559953]
+AltSum: A000166 [1, 0, 1, -2, 9, -44, 265, -1854, 14833, -133496]
+DiagSum: A122852 [1, 1, 2, 3, 6, 11, 24, 51, 122, 291]
+Central: A001813 [1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600]
+LeftSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+RightSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+PosHalf: A010842 [1, 3, 10, 38, 168, 872, 5296, 37200, 297856, 2681216]
+NegHalf: A000023 [1, -1, 2, -2, 8, 8, 112, 656, 5504, 49024]
+TransUnos: A000522 [1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410]
+TransAlts: A000166 [1, 0, 1, -2, 9, -44, 265, -1854, 14833, -133496]
+TransSqrs: [ Info: Dowloaded 0,1,10,81,65.json to C:\Users\User\GitHub2021\IntegerTriangles.jl\data
+nothing [0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056, 64116657]
+TransNat0: A093964 [0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281]
+TransNat1: A001339 [1, 3, 11, 49, 261, 1631, 11743, 95901, 876809, 8877691]
+PolyVal2: A010844 [1, 3, 13, 79, 633, 6331, 75973, 1063623, 17017969, 306323443]
+PolyVal3: A010845 [1, 4, 25, 226, 2713, 40696, 732529, 15383110, 369194641, 9968255308]
+Main.TriangleTraitCard
+
diff --git a/data/profile.txt b/data/profile.txt
index 3be8098..9929060 100644
--- a/data/profile.txt
+++ b/data/profile.txt
@@ -13,3 +13,78 @@
│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
│ nothing │ Laguerre │ Rev │ TransNat1 │ 1, 3, 15, 97, 753, 6771, 68983, 783945 │
└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
+┌──────────┬────────────┬──────┬───────────┬─────────────────────────────────────────────┐
+│ A-number │ Triangle │ Form │ Function │ Sequence │
+├──────────┼────────────┼──────┼───────────┼─────────────────────────────────────────────┤
+│ A000302 │ Binomial │ Std │ PolyVal3 │ 1, 4, 16, 64, 256, 1024, 4096, 16384 │
+│ A001333 │ SchroederB │ Inv │ AltSum │ 1, -1, 3, -7, 17, -41, 99, -239 │
+│ A006012 │ SchroederL │ Inv │ AltSum │ 1, -2, 6, -20, 68, -232, 792, -2704 │
+│ A026302 │ Motzkin │ Rev │ Central │ 1, 2, 9, 44, 230, 1242, 6853, 38376 │
+│ A025167 │ Laguerre │ Std │ PosHalf │ 1, 3, 17, 139, 1473, 19091, 291793, 5129307 │
+│ A103194 │ Laguerre │ Std │ TransNat0 │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ C000262 │ Laguerre │ Std │ TransNat1 │ 1, 3, 13, 73, 501, 4051, 37633, 394353 │
+│ A103194 │ Lah │ Std │ TransSqrs │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ nothing │ Laguerre │ Rev │ TransNat1 │ 15, 97, 753, 6771, 68983, 783945 │
+└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
+┌──────────┬────────────┬──────┬───────────┬─────────────────────────────────────────────┐
+│ A-number │ Triangle │ Form │ Function │ Sequence │
+├──────────┼────────────┼──────┼───────────┼─────────────────────────────────────────────┤
+│ A000302 │ Binomial │ Std │ PolyVal3 │ 1, 4, 16, 64, 256, 1024, 4096, 16384 │
+│ A001333 │ SchroederB │ Inv │ AltSum │ 1, -1, 3, -7, 17, -41, 99, -239 │
+│ A006012 │ SchroederL │ Inv │ AltSum │ 1, -2, 6, -20, 68, -232, 792, -2704 │
+│ A026302 │ Motzkin │ Rev │ Central │ 1, 2, 9, 44, 230, 1242, 6853, 38376 │
+│ A025167 │ Laguerre │ Std │ PosHalf │ 1, 3, 17, 139, 1473, 19091, 291793, 5129307 │
+│ A103194 │ Laguerre │ Std │ TransNat0 │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ C000262 │ Laguerre │ Std │ TransNat1 │ 1, 3, 13, 73, 501, 4051, 37633, 394353 │
+│ A103194 │ Lah │ Std │ TransSqrs │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ nothing │ Laguerre │ Rev │ TransNat1 │ 1, 3, 15, 97, 753, 6771, 68983, 783945 │
+└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
+┌──────────┬────────────┬──────┬───────────┬─────────────────────────────────────────────┐
+│ A-number │ Triangle │ Form │ Function │ Sequence │
+├──────────┼────────────┼──────┼───────────┼─────────────────────────────────────────────┤
+│ A000302 │ Binomial │ Std │ PolyVal3 │ 1, 4, 16, 64, 256, 1024, 4096, 16384 │
+│ A001333 │ SchroederB │ Inv │ AltSum │ 1, -1, 3, -7, 17, -41, 99, -239 │
+│ A006012 │ SchroederL │ Inv │ AltSum │ 1, -2, 6, -20, 68, -232, 792, -2704 │
+│ A026302 │ Motzkin │ Rev │ Central │ 1, 2, 9, 44, 230, 1242, 6853, 38376 │
+│ A025167 │ Laguerre │ Std │ PosHalf │ 1, 3, 17, 139, 1473, 19091, 291793, 5129307 │
+│ A103194 │ Laguerre │ Std │ TransNat0 │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ C000262 │ Laguerre │ Std │ TransNat1 │ 1, 3, 13, 73, 501, 4051, 37633, 394353 │
+│ A103194 │ Lah │ Std │ TransSqrs │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ nothing │ Laguerre │ Rev │ TransNat1 │ 1, 3, 15, 97, 753, 6771, 68983, 783945 │
+└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
+┌──────────┬────────────┬──────┬───────────┬─────────────────────────────────────────────┐
+│ A-number │ Triangle │ Form │ Function │ Sequence │
+├──────────┼────────────┼──────┼───────────┼─────────────────────────────────────────────┤
+│ A000302 │ Binomial │ Std │ PolyVal3 │ 1, 4, 16, 64, 256, 1024, 4096, 16384 │
+│ A001333 │ SchroederB │ Inv │ AltSum │ 1, -1, 3, -7, 17, -41, 99, -239 │
+│ A006012 │ SchroederL │ Inv │ AltSum │ 1, -2, 6, -20, 68, -232, 792, -2704 │
+│ A026302 │ Motzkin │ Rev │ Central │ 1, 2, 9, 44, 230, 1242, 6853, 38376 │
+│ A025167 │ Laguerre │ Std │ PosHalf │ 1, 3, 17, 139, 1473, 19091, 291793, 5129307 │
+│ A103194 │ Laguerre │ Std │ TransNat0 │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ C000262 │ Laguerre │ Std │ TransNat1 │ 1, 3, 13, 73, 501, 4051, 37633, 394353 │
+│ A103194 │ Lah │ Std │ TransSqrs │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ nothing │ Laguerre │ Rev │ TransNat1 │ 1, 3, 15, 97, 753, 6771, 68983, 783945 │
+└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
+┌──────────┬────────────┬──────┬───────────┬─────────────────────────────────────────────┐
+│ A-number │ Triangle │ Form │ Function │ Sequence │
+├──────────┼────────────┼──────┼───────────┼─────────────────────────────────────────────┤
+│ A000302 │ Binomial │ Std │ PolyVal3 │ 1, 4, 16, 64, 256, 1024, 4096, 16384 │
+│ A001333 │ SchroederB │ Inv │ AltSum │ 1, -1, 3, -7, 17, -41, 99, -239 │
+│ A006012 │ SchroederL │ Inv │ AltSum │ 1, -2, 6, -20, 68, -232, 792, -2704 │
+│ A026302 │ Motzkin │ Rev │ Central │ 1, 2, 9, 44, 230, 1242, 6853, 38376 │
+│ A025167 │ Laguerre │ Std │ PosHalf │ 1, 3, 17, 139, 1473, 19091, 291793, 5129307 │
+│ A103194 │ Laguerre │ Std │ TransNat0 │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ C000262 │ Laguerre │ Std │ TransNat1 │ 1, 3, 13, 73, 501, 4051, 37633, 394353 │
+│ A103194 │ Lah │ Std │ TransSqrs │ 0, 1, 6, 39, 292, 2505, 24306, 263431 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ A111884 │ Lah │ Std │ TransAlts │ 1, -1, -1, -1, 1, 19, 151, 1091 │
+│ nothing │ Laguerre │ Rev │ TransNat1 │ 1, 3, 15, 97, 753, 6771, 68983, 783945 │
+└──────────┴────────────┴──────┴───────────┴─────────────────────────────────────────────┘
diff --git a/data/stripped b/data/stripped
index c810179..50aa0ca 100644
--- a/data/stripped
+++ b/data/stripped
@@ -1,5 +1,5 @@
# OEIS Sequence Data (http://oeis.org/stripped.gz)
-# Last Modified: April 1 03:24 UTC 2021
+# Last Modified: April 19 04:18 UTC 2021
# Use of this content is governed by the
# OEIS End-User License: http://oeis.org/LICENSE
A000001 ,0,1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,1,2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,13,2,2,1,13,1,2,4,267,1,4,1,5,1,4,1,50,1,2,3,4,1,6,1,52,15,2,1,15,1,2,1,12,1,10,1,4,2,
@@ -5332,7 +5332,7 @@ A005327 ,1,0,1,6,91,2820,177661,22562946,5753551231,2940064679040,30076861666579
A005328 ,1,3,28,510,18631,1351413,194192398,55272612720,31184369778961,34909296450535023,77616063417393956368,343049603717222441078130,3016429354620114423122804491,52801416275268069417410827891833,
A005329 ,1,1,3,21,315,9765,615195,78129765,19923090075,10180699028325,10414855105976475,21319208401933844325,87302158405919092510875,715091979502883286756577125,11715351900195736886933003038875,383876935713713710574133710574817125,
A005330 ,1,5,40,644,21496,1471460,204062440,56865072164,31688930152696,35223651007587140,78001790003385408040,343983307379873262633284,3020895063527811952260491896,52843677532033943174017588842020,1841795434229559227318546660111716840,
-A005331 ,1,2,5,20,179,4082,218225,25316720,6135834479,3047003143022,3067545380897645,6223557209578656620,25360384878802358268779,207167485813280961035481962,3389045635023473628621934703465,
+A005331 ,1,2,5,20,179,4082,218225,25316720,6135834479,3047003143022,3067545380897645,6223557209578656620,25360384878802358268779,207167485813280961035481962,3389045635023473628621934703465,110960673463328354866093662279119720,7268699514166911556909103208203294815079,
A005332 ,1,7,58,838,25171,1610977,214838128,58540023808,32208188445841,35543124039418147,78391002506394742198,344921660620756227029578,3025372940760065880037836511,52886001393832278158415800800117,
A005333 ,1,5,205,36317,23679901,56294206205,502757743028605,17309316971673776957,2333508400614646874734621,1243000239291173897659593056765,2629967962392578020413552363565293565,22170252073745058975210005804934596601690557,
A005334 ,1,1,34,7037,6317926,21073662977,251973418941994,10878710974408306717,1727230695707098000548430,1028983422758641650604161840065,2342608062302306704492272616530549874,20683716767972841770515007707311751484424893,
@@ -6409,32 +6409,32 @@ A006404 ,1,1,2,5,10,29,96,339,1320,5473,
A006405 ,1,1,2,5,9,24,70,222,785,3055,
A006406 ,1,1,2,4,9,24,81,274,1071,4357,18416,80040,356109,1610910,7399114,
A006407 ,1,1,2,4,8,20,58,177,630,2410,9772,41423,181586,814412,3722445,
-A006408 ,4,39,190,651,1792,4242,8988,17490,31812,
-A006409 ,10,190,1568,8344,33580,111100,317680,811096,
-A006410 ,20,651,8344,64667,361884,1607125,5997992,
-A006411 ,3,20,75,210,490,1008,1890,3300,5445,8580,13013,19110,27300,38080,52020,69768,92055,119700,153615,194810,244398,303600,373750,456300,552825,665028,794745,943950,1114760,1309440,
-A006412 ,4,75,604,3150,12480,40788,115500,292578,677820,1459315,
-A006413 ,5,210,3150,27556,170793,829920,3359356,11786190,36845718,
+A006408 ,4,39,190,651,1792,4242,8988,17490,31812,54769,90090,142597,218400,325108,472056,670548,934116,1278795,1723414,2289903,3003616,3893670,4993300,6340230,7977060,9951669,12317634,15134665,18469056,22394152,26990832,32348008,38563140,
+A006409 ,10,190,1568,8344,33580,111100,317680,811096,1891318,4094090,8328320,16071120,29636984,52540472,89974880,149432720,241497410,380839382,587453856,888181800,1318560100,1925051700,2767711440,3923348520,5489251950,7587551010,10370288640,
+A006410 ,20,651,8344,64667,361884,1607125,5997992,19535997,57014776,151986562,375470160,869285378,1902886024,3966657702,7920130544,15220758070,28268206764,50910912965,89176474920,152305796565,254193384900,415363487955,665644575960,1047743815755,
+A006411 ,3,20,75,210,490,1008,1890,3300,5445,8580,13013,19110,27300,38080,52020,69768,92055,119700,153615,194810,244398,303600,373750,456300,552825,665028,794745,943950,1114760,1309440,1530408,1780240,2061675,2377620,2731155,3125538,
+A006412 ,4,75,604,3150,12480,40788,115500,292578,677820,1459315,2954952,5679700,10438272,18449760,31511880,52213596,84206100,132543411,204105220,308116050,456776320,666022500,956435220,1354315950,1892954700,2614113099,3569749200,4824012424,
+A006413 ,5,210,3150,27556,170793,829920,3359356,11786190,36845718,104719524,274707420,672982128,1554007910,3407724936,7139933088,14366348780,27878652291,52364814150,95497666810,169546939380,293722986375,497527759560,825473130300,1343631834090,
A006414 ,1,9,40,125,315,686,1344,2430,4125,6655,10296,15379,22295,31500,43520,58956,78489,102885,133000,169785,214291,267674,331200,406250,494325,597051,716184,853615,1011375,1191640,1396736,1629144,1891505,2186625,2517480,2887221,
A006415 ,4,104,1020,6092,26670,94128,283338,754380,1821534,4061200,
A006416 ,1,8,20,38,63,96,138,190,253,328,416,518,635,768,918,1086,1273,1480,1708,1958,2231,2528,2850,3198,3573,3976,4408,4870,5363,5888,6446,7038,7665,8328,9028,9766,10543,11360,12218,13118,14061,15048,
-A006417 ,1,20,131,469,1262,2862,5780,10725,18647,30784,48713,74405,
-A006418 ,1,38,469,3008,12843,42602,119042,293578,658021,1367170,2670203,
+A006417 ,1,20,131,469,1262,2862,5780,10725,18647,30784,48713,74405,110284,159290,224946,311429,423645,567308,749023,976373,1258010,1603750,2024672,2533221,3143315,3870456,4731845,5746501,6935384,8321522,9930142,11788805,13927545,16379012,
+A006418 ,1,38,469,3008,12843,42602,119042,293578,658021,1367170,2670203,4953136,8794967,15040494,24893192,40031954,62755945,96162286,144361777,212738384,308258755,439838594,618773310,859240970,1178886221,1599494506,2147766583,2856204064,
A006419 ,0,1,7,37,176,794,3473,14893,63004,263950,1097790,4540386,18696432,76717268,313889477,1281220733,5219170052,21224674118,86188320962,349550141078,1416102710912,5731427140268,23177285611082,93655986978002,378195990166136,1526289367335244,
-A006420 ,1,16,150,1104,7077,41504,228810,1205520,6135690,30391520,147277676,700990752,3286733805,15215673408,
-A006421 ,1,30,449,4795,41850,319320,2213665,14283280,87169790,508887860,2865204762,15654301865,
-A006422 ,4,47,240,831,2282,5362,11256,21690,39072,66649,
-A006423 ,10,240,2246,12656,52164,173776,495820,1256992,2902702,
-A006424 ,20,831,12656,109075,648792,2978245,11293436,36973989,
-A006425 ,4,79,900,7885,59080,398846,2499096,14805705,83969600,459868530,
-A006426 ,10,340,5846,71372,706068,6052840,46759630,333746556,2238411692,
-A006427 ,20,1071,26320,431739,5494896,58677420,550712668,4681144391,
-A006428 ,0,3,36,135,360,798,1568,2826,4770,7645,11748,17433,
-A006429 ,0,4,135,1368,7350,28400,89073,241220,585057,1301420,2699125,
-A006430 ,0,5,360,7350,73700,474588,2292790,9046807,30676440,92393015,
+A006420 ,1,16,150,1104,7077,41504,228810,1205520,6135690,30391520,147277676,700990752,3286733805,15215673408,69675615234,316058238864,1421891923038,6350464644960,28179908990772,124327908683616,545691921346146,2383936774151616,10370479696102500,
+A006421 ,1,30,449,4795,41850,319320,2213665,14283280,87169790,508887860,2865204762,15654301865,83388235348,434685964540,2223970137825,11194499812388,55546566721430,272142754971892,1318317357277470,6321681903231990,30037740651227756,141545610360126400,
+A006422 ,4,47,240,831,2282,5362,11256,21690,39072,66649,108680,170625,259350,383348,552976,780708,1081404,1472595,1974784,2611763,3410946,4403718,5625800,7117630,8924760,11098269,13695192,16778965,20419886,24695592,29691552,35501576,
+A006423 ,10,240,2246,12656,52164,173776,495820,1256992,2902702,6214208,12494482,23827440,43430088,76120288,128926232,211867328,338940050,529346384,809006814,1212404336,1784810764,2584951600,3688170980,5190163680,7211346870,9901950240,13447909290,
+A006424 ,20,831,12656,109075,648792,2978245,11293436,36973989,107727724,285451894,699013380,1601397330,3465135024,7135903782,14072047976,26707904230,48991682628,87164772761,150869282184,254695011933,420306632200,679327313795,1077197343300,1678276223715,
+A006425 ,4,79,900,7885,59080,398846,2499096,14805705,83969600,459868530,2447439384,12718070274,64766697520,324156347260,1598200903280,7776728909121,37404399901296,178060831286890,839857764202520,3928581810398630,18239060530882224,84101317494787684,
+A006426 ,10,340,5846,71372,706068,6052840,46759630,333746556,2238411692,14277544216,87376309020,516495616120,2964332933800,16586670357200,90782049175614,487329793111260,2571575908919740,13364166071956280,68507393061864020,346874109053120616,
+A006427 ,20,1071,26320,431739,5494896,58677420,550712668,4681144391,36786186216,271221867098,1896796135920,12688048319278,81709791432384,509222462694582,3083998029716868,18213177504318335,105186858991413976,595499805083872458,3311524095424508480,
+A006428 ,0,3,36,135,360,798,1568,2826,4770,7645,11748,17433,25116,35280,48480,65348,86598,113031,145540,185115,232848,289938,357696,437550,531050,639873,765828,910861,1077060,1266660,1482048,1725768,2000526,2309195,2654820,3040623,
+A006429 ,0,4,135,1368,7350,28400,89073,241220,585057,1301420,2699125,5282172,9842430,17584416,30289835,50530680,81940901,129557940,200246795,303220720,450674190,658545360,947426925,1343646044,1880535825,2599922780,3553856649,4806611060,
+A006430 ,0,5,360,7350,73700,474588,2292790,9046807,30676440,92393015,252872984,639382605,1512137536,3377126024,7176513960,14599539314,28575632350,54036739617,99069119952,176618150000,306965183268,521265871700,866527603370,1412513294049,
A006431 ,0,1,2,3,5,6,7,8,11,14,15,23,24,32,56,96,128,224,384,512,896,1536,2048,3584,6144,8192,14336,24576,32768,57344,98304,131072,229376,393216,524288,917504,1572864,2097152,3670016,6291456,8388608,14680064,
-A006432 ,0,3,60,650,5352,37681,239752,1421226,7996160,43219990,226309800,1154900708,
-A006433 ,0,4,175,3324,42469,429120,3711027,28723640,204598130,1366223880,8664086470,
+A006432 ,0,3,60,650,5352,37681,239752,1421226,7996160,43219990,226309800,1154900708,5769562736,28312118565,136830224464,652656300122,3077631550512,14367512295274,66478236840680,305161336656876,1390869368495728,6298727501142218,28358908010334960,
+A006433 ,0,4,175,3324,42469,429120,3711027,28723640,204598130,1366223880,8664086470,52673351080,309164754285,1761471681568,9783594370723,53154274959360,283267669144390,1484104565936920,7658877239935362,38993558097982312,196127054929939810,
A006434 ,4,120,1230,7424,32424,113584,338742,893220,2136618,4721728,9770904,
A006435 ,10,705,14478,154420,1092640,5826492,25240410,93203561,303143970,889015725,
A006436 ,10,192,1630,8924,36834,124560,362934,941820,2227368,4881448,889015725,
@@ -6469,10 +6469,10 @@ A006464 ,0,3,6,4,4,2,4,6,4,2,6,4,2,4,4,6,4,2,6,4,4,2,4,6,2,4,6,4,2,4,4,6,4,2,6,4
A006465 ,324,63,1,1023,64,1023,1,63,1023,1,63,1023,1,62,1,1023,63,1,1023,64,1023,1,63,1023,1,62,1,1023,64,1023,1,63,1023,1,62,1,1023,63,1,1023,63,1,1023,64,1023,1,
A006466 ,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2,
A006467 ,0,4,3,1,3,5,1,3,5,3,3,1,5,3,1,3,3,5,3,1,3,5,1,3,3,5,3,1,5,3,1,3,5,3,3,1,3,5,1,3,5,3,3,1,5,3,1,3,5,3,3,1,3,5,1,3,3,5,3,1,5,3,1,3,3,5,3,1,3,5,1,3,5,3,3,1,5,3,1,3,3,5,3,1,3,5,1,3,3,5,3,1,5,3,1,3,3,5,3,1,3,5,1,3,5,
-A006468 ,5,37,150,449,1113,2422,4788,8790,15213,25091,39754,60879,90545,131292,
+A006468 ,5,37,150,449,1113,2422,4788,8790,15213,25091,39754,60879,90545,131292,186184,258876,353685,475665,630686,825517,1067913,1366706,1731900,2174770,2707965,3345615,4103442,4998875,6051169,7281528,8713232,10371768,12284965,14483133,16999206,
A006469 ,10,79,340,1071,2772,6258,12768,24090,42702,71929,116116,180817,273000,401268,576096,810084,1118226,1518195,2030644,2679523,3492412,4500870,5740800,7252830,9082710,11281725,13907124,17022565,20698576,25013032,30051648,35908488,
A006470 ,2,15,60,175,420,882,1680,2970,4950,7865,12012,17745,25480,35700,48960,65892,87210,113715,146300,185955,233772,290950,358800,438750,532350,641277,767340,912485,1078800,1268520,1484032,1727880,2002770,2311575,2657340,3043287,3472820,3949530,4477200,5059810,5701542,6406785,7180140,
-A006471 ,5,84,650,3324,13020,42240,118998,300300,693693,1490060,3011580,
+A006471 ,5,84,650,3324,13020,42240,118998,300300,693693,1490060,3011580,5779592,10608000,18728832,31957620,52907400,85261341,134115300,206402966,311417700,461446700,672534720,965396250,1366496820,1909325925,2635885980,3598423704,4861432400,6503955744,8622225920,
A006472 ,1,1,3,18,180,2700,56700,1587600,57153600,2571912000,141455160000,9336040560000,728211163680000,66267215894880000,6958057668962400000,834966920275488000000,113555501157466368000000,17373991677092354304000000,2970952576782792585984000000,
A006473 ,1,30,30240,1816214400,10137091700736000,7561714896123855667200000,1025113885554181044609786839040000000,32964677266721834921175915315161407370035200000000,318071672921132854486459356650996997744817246158245068800000000000,
A006474 ,1,2,4,9,16,20,30,42,49,64,
@@ -8678,7 +8678,7 @@ A008673 ,1,1,1,2,2,3,4,5,6,7,9,10,12,14,16,19,21,24,27,30,34,38,42,46,51,56,61,6
A008674 ,1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58,65,74,82,91,102,113,124,138,151,165,182,198,216,236,256,277,301,325,350,379,407,437,471,504,539,578,617,658,703,748,795,847,899,953,1012,1071,1133,1200,1267,1337,1413,1489,1568,1653,
A008675 ,1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227,252,278,307,339,372,408,448,489,534,583,634,689,749,811,878,950,1025,1106,1192,1282,1378,1481,1588,1702,1823,1949,2083,
A008676 ,1,0,0,1,0,1,1,0,1,1,1,1,1,1,1,2,1,1,2,1,2,2,1,2,2,2,2,2,2,2,3,2,2,3,2,3,3,2,3,3,3,3,3,3,3,4,3,3,4,3,4,4,3,4,4,4,4,4,4,4,5,4,4,5,4,5,5,4,5,5,5,5,5,5,5,6,5,5,
-A008677 ,1,0,0,1,0,1,1,1,1,1,2,1,2,2,2,3,2,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,8,8,9,9,9,10,10,11,11,12,12,12,14,13,14,15,15,16,16,17,17,18,19,19,20,20,21,22,22,23,24,24,25,
+A008677 ,1,0,0,1,0,1,1,1,1,1,2,1,2,2,2,3,2,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,8,8,9,9,9,10,10,11,11,12,12,12,14,13,14,15,15,16,16,17,17,18,19,19,20,20,21,22,22,23,24,24,25,26,26,27,28,29,29,30,31,31,33,33,34,35,35,37,37,38,39,40,41,41,43,
A008678 ,1,0,0,1,0,1,1,1,1,2,2,1,3,2,3,4,3,4,5,5,5,7,6,7,9,8,9,11,11,11,14,13,14,17,16,18,20,20,21,24,24,25,29,28,30,34,33,35,39,39,41,45,45,47,52,52,54,59,59,62,67,67,
A008679 ,1,0,0,1,1,0,1,1,1,1,1,1,2,1,1,2,2,1,2,2,2,2,2,2,3,2,2,3,3,2,3,3,3,3,3,3,4,3,3,4,4,3,4,4,4,4,4,4,5,4,4,5,5,4,5,5,5,5,5,5,6,5,5,6,6,5,6,6,6,6,6,6,7,6,6,7,7,6,7,7,7,7,7,7,8,7,
A008680 ,1,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,5,5,6,6,6,7,8,8,8,9,10,10,11,11,12,13,13,14,15,15,16,17,18,18,19,20,21,22,22,23,25,25,26,27,28,29,30,31,32,33,34,35,37,37,38,40,41,42,43,44,46,47,48,49,51,52,53,55,56,57,
@@ -11758,7 +11758,7 @@ A011753 ,11,20,30,420,5460,60060,746130,80059980,900029130,10407767370,110086937
A011754 ,1,2,2,4,3,6,6,5,6,8,9,13,10,11,14,15,11,14,14,17,17,20,19,22,16,18,24,30,25,25,25,26,26,34,29,32,27,34,36,32,28,39,38,39,34,34,45,38,41,33,41,46,42,35,39,42,39,40,42,48,56,56,49,57,56,51,45,47,55,55,64,68,58,
A011755 ,1,3,9,17,37,49,91,123,177,217,327,375,531,615,735,863,1135,1243,1585,1745,1997,2217,2723,2915,3415,3727,4213,4549,5361,5601,6531,7043,7703,8247,9087,9519,10851,11535,12471,13111,14751,15255,17061,17941,19021,20033,
A011756 ,2,5,13,29,47,73,107,151,197,257,317,397,467,571,659,769,883,1019,1151,1291,1453,1607,1783,1987,2153,2371,2593,2791,3037,3307,3541,3797,4073,4357,4657,4973,5303,5641,5939,6301,6679,7019,7477,
-A011757 ,2,7,23,53,97,151,227,311,419,541,661,827,1009,1193,1427,1619,1879,2143,2437,2741,3083,3461,3803,4211,4637,5051,5519,6007,6481,6997,7573,8161,8737,9341,9931,10627,11321,12049,12743,13499,14327,
+A011757 ,2,7,23,53,97,151,227,311,419,541,661,827,1009,1193,1427,1619,1879,2143,2437,2741,3083,3461,3803,4211,4637,5051,5519,6007,6481,6997,7573,8161,8737,9341,9931,10627,11321,12049,12743,13499,14327,15101,15877,16747,17609,18461,
A011758 ,1,1,1,1,1,-1,-1,1,1,-1,1,-1,1,
A011759 ,0,0,0,0,0,1,1,0,0,1,0,1,0,
A011760 ,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,
@@ -14343,7 +14343,7 @@ A014338 ,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
A014339 ,1,1,0,-1,-1,-1,0,1,1,0,0,-1,-1,0,1,1,1,0,-1,-1,0,0,1,1,0,-1,-1,-1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,-1,-1,-1,0,1,1,0,0,-1,-1,0,1,1,1,0,-1,-1,0,0,1,1,0,-1,
A014340 ,0,0,0,0,24,240,2040,15120,106680,726768,4861560,32126160,210749880,1375923120,8955795576,58177326480,377439735480,2446686537840,15851581501560,102662600100432,664738663537080,
A014341 ,0,0,0,0,1,10,85,630,4445,30282,202565,1338590,8781245,57330130,373158149,2424055270,15726655645,101945272410,660482562565,4277608337518,27697444314045,179313224367970,
-A014342 ,4,12,29,58,111,188,305,462,679,968,1337,1806,2391,3104,3953,4978,6175,7568,9185,11030,13143,15516,18177,21150,24471,28152,32197,36678,41543,46828,52621,58874,65659,73000,80949,89462,98631,108396,118869,130102,
+A014342 ,4,12,29,58,111,188,305,462,679,968,1337,1806,2391,3104,3953,4978,6175,7568,9185,11030,13143,15516,18177,21150,24471,28152,32197,36678,41543,46828,52621,58874,65659,73000,80949,89462,98631,108396,118869,130102,142071,
A014343 ,8,36,114,291,669,1386,2678,4851,8373,13858,22134,34263,51635,75972,109374,154483,214383,292812,394148,523521,686901,891112,1143936,1454187,1831973,2288400,2836044,3488969,4262541,5173836,6241612,7486437,8930649,10598848,
A014344 ,16,96,376,1160,3121,7532,16754,34796,68339,127952,229956,398688,669781,1094076,1742710,2713604,4139111,6195712,9115304,13199072,18833449,26509260,36843322,50603884,68740107,92414192,123039628,162323200,212312453,275448380,
A014345 ,4,12,38,118,362,1082,3166,8910,24426,64226,165262,413418,1021362,2490686,6009150,14401410,34098042,80281962,187356750,432549154,992941250,2256712462,5088826238,11408805862,25425739346,56383362854,124565557898,274390550594,
@@ -24451,7 +24451,7 @@ A024446 ,1,31,638,10982,171171,2506917,35201776,479688604,6392929061,83765551883
A024447 ,0,6,31,101,288,652,1349,2451,4222,7122,11121,17041,25118,35352,48559,65943,88422,115262,148829,189157,235804,292052,357705,435491,528902,635962,755545,890793,1040232,1207472,1409783,1635103,1888690,2165022,2481945,
A024448 ,30,247,1358,5102,16186,41817,98190,220628,441410,852887,1551568,2631642,4293186,6866813,10757450,16151192,23873746,34440605,48249066,66877582,91117898,122953643,165196270,218615372,284119458,364962773,462059210,579605426,732954370,
A024449 ,210,2927,20581,107315,414849,1376640,4224150,11063618,27395788,62364155,129081579,252768753,480307611,885449578,1541654028,2623783892,4318819858,6832984023,10644660237,16195499543,24304992465,36231495836,52916319106,75433702422,
-A024450 ,4,13,38,87,208,377,666,1027,1556,2397,3358,4727,6408,8257,10466,13275,16756,20477,24966,30007,35336,41577,48466,56387,65796,75997,86606,98055,109936,122705,138834,155995,174764,194085,216286,239087,263736,290305,
+A024450 ,4,13,38,87,208,377,666,1027,1556,2397,3358,4727,6408,8257,10466,13275,16756,20477,24966,30007,35336,41577,48466,56387,65796,75997,86606,98055,109936,122705,138834,155995,174764,194085,216286,239087,263736,290305,318194,
A024451 ,0,1,5,31,247,2927,40361,716167,14117683,334406399,9920878441,314016924901,11819186711467,492007393304957,21460568175640361,1021729465586766997,54766551458687142251,3263815694539731437539,201015517717077830328949,13585328068403621603022853,
A024452 ,1,3,5,10,15,23,31,42,55,69,86,105,125,148,173,200,230,262,296,331,369,409,452,498,547,597,649,702,757,819,883,950,1017,1090,1164,1240,1320,1402,1487,1574,1663,1757,1851,1948,
A024453 ,3,14,48,124,279,543,981,1710,2758,4329,6519,9365,13088,18023,24448,32237,42031,53897,67765,84548,104253,127677,155845,188299,224778,266201,312202,363845,426136,495751,574268,660165,758682,865898,984968,1116797,
@@ -27438,7 +27438,7 @@ A027433 ,0,2,18,116,646,3324,16302,77356,358424,1630988,7317424,32458400,1426385
A027434 ,2,3,4,4,5,5,6,6,6,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,17,17,17,17,18,18,18,18,
A027435 ,1,2,4,6,10,11,17,21,27,29,39,42,54,57,62,70,86,89,107,113,120,125,147,152,172,178,196,204,232,236,266,282,294,302,320,329,365,374,388,400,440,446,488,501,518,529,575,586,628,638,657,672,724,733,758,778,
A027436 ,0,1,2,-4,16,-80,432,-2304,10944,-35328,-74112,2736384,-30853632,238663680,-1247457280,2201247744,32530722816,-320650199040,156266184704,18314630348800,-20667999748096,-3428200020508672,
-A027437 ,2,5,10,21,43,86,173,347,695,1391,2783,5567,11134,22268,44536,89072,178145,356290,712581,1425162,2850325,5700650,11401300,22802600,45605201,91210402,182420805,364841611,
+A027437 ,2,5,10,21,43,86,173,347,695,1391,2783,5567,11134,22268,44536,89072,178145,356290,712581,1425162,2850325,5700650,11401300,22802600,45605201,91210402,182420805,364841611,729683222,1459366444,2918732888,5837465777,11674931554,
A027438 ,2,5,5,7,43,43,173,347,139,107,23,293,293,293,293,293,79,79,883,883,114013,114013,114013,114013,1232573,1232573,1499,364841611,364841611,364841611,364841611,170801,170801,56812319,56812319,56812319,56812319,
A027439 ,1,4,11,27,65,158,388,957,2362,5827,14369,35427,87344,215348,530953,1309104,3227695,7958119,19621313,48377742,119278760,294090269,725100502,1787786943,4407916069,10868031067,
A027440 ,1,3,4,6,8,10,12,
@@ -27837,7 +27837,7 @@ A027832 ,1,5,14,315,2634,301262,8035168,4451407563,392447922178,1028823851939030
A027833 ,1,2,2,3,3,4,3,6,2,5,2,6,2,2,4,3,5,3,4,5,12,2,6,9,6,5,4,3,4,20,2,2,4,4,19,2,3,2,4,8,11,5,3,3,3,10,5,4,2,17,3,6,3,3,9,9,2,6,2,6,5,6,2,3,2,3,9,4,7,3,7,20,4,7,6,5,3,7,3,20,2,14,4,10,2,3,6,4,2,2,7,2,6,3,
A027834 ,1,9,296,20958,2554344,474099840,124074010080,43429847756400,19565965561887360,11018376449767451520,7579467449864423769600,6251471405353507523097600,6087988343847192559805952000,6910412728595671664966422425600,9042510998634333921282477985689600,
A027835 ,1,6,52,892,21291,658885,24617866,1077142765,53918557215,3036369842197,189881640057942,13051044976503663,977672716919010876,79267586388173032966,6914956215333832011058,645771787789692953182732,64277686448923785217048191,6793045601578652098886514581,759656437858515775195264228768,89619947709601175930862298926038,
-A027836 ,1,2,8,43,268,1824,13156,98865,765948,6075256,49094708,402801425,3346590068,28099903160,238079915640,2032914717645,17476713955548,151143219598008,1314045772469632,11478299163026540,100688538612524720,
+A027836 ,1,2,8,43,268,1824,13156,98865,765948,6075256,49094708,402801425,3346590068,28099903160,238079915640,2032914717645,17476713955548,151143219598008,1314045772469632,11478299163026540,100688538612524720,886622619082002120,7834289222109530340,
A027837 ,1,7,97,2143,68641,3011263,173773153,12785668351,1169623688353,130305512589247,17376934722756577,2733655173624167551,501034099176714373921,105847486567006696384831,
A027838 ,1,7,22,111,486,3772,29142,275871,2830459,32028882,392744078,5201524044,73943424582,1123603726896,18176728661832,311951284854975,5661698774848910,108355867352001811,2181096921557783966,46066653293892718506,1018705098450761473704,
A027839 ,1,15,124,2431,68766,3025596,173773496,12786773247,1169623901221,130305653188890,17376934722757908,2733655198336260124,501034099176714376118,105847486572700895182728,25534322201330399433420024,6976464857439636995547805183,
@@ -33511,7 +33511,7 @@ A033506 ,1,3,22,131,823,5096,31687,196785,1222550,7594361,47177097,293066688,182
A033507 ,1,5,71,823,10012,120465,1453535,17525619,211351945,2548684656,30734932553,370635224561,4469527322891,53898461609719,649966808093412,7838012982224913,94519361817920403,1139818186429110279,13745178487929574337,165754445655292452448,
A033508 ,1,8,228,5096,120465,2810694,65805403,1539222016,36012826776,842518533590,19711134149599,461148537211748,10788744980331535,252406631116215534,5905146419664967132,138153075553825008696,
A033509 ,1,13,733,31687,1453535,65805403,2989126727,135658637925,6158217253688,279533139565077,12688781322524383,575975678462394151,26145024935911561519,1186789728933332428003,53871436268769248658909,
-A033510 ,1,21,2356,196785,17525619,1539222016,135658637925,11945257052321,1052091957273408,92657526436631289,8160498611028648795,718704019165239462736,63297158846544276862187,
+A033510 ,1,21,2356,196785,17525619,1539222016,135658637925,11945257052321,1052091957273408,92657526436631289,8160498611028648795,718704019165239462736,63297158846544276862187,5574656798624151746571203,490966794038702258639391536,43240039834820302340627322251,
A033511 ,1,34,7573,1222550,211351945,36012826776,6158217253688,1052091957273408,179788343101980135,30721240815429999078,5249581929453966097649,897032469743945346623442,153282416794739031814924079,26192460807219455656314349664,
A033512 ,1,55,24342,7594361,2548684656,842518533590,279533139565077,92657526436631289,30721240815429999078,10185111919160666118608,3376771882017849561260401,1119528305492897225312120742,371166848426649070379972358241,123056076089539426001744759117065,
A033513 ,1,89,78243,47177097,30734932553,19711134149599,12688781322524383,8160498611028648795,5249581929453966097649,3376771882017849561260401,2172138783673094193937750015,1397239543795157686038029989037,898783117890438425073750503319527,
@@ -33641,14 +33641,14 @@ A033636 ,1,21,1025310,393143628567690,4161601591128480023923529880,3025979870886
A033637 ,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,21,22,24,25,27,28,30,32,33,35,36,40,42,44,45,48,49,50,54,55,56,60,63,64,66,70,72,75,77,80,81,84,88,90,96,98,99,100,101,105,108,110,112,120,121,125,126,128,132,135,140,
A033638 ,1,1,2,3,5,7,10,13,17,21,26,31,37,43,50,57,65,73,82,91,101,111,122,133,145,157,170,183,197,211,226,241,257,273,290,307,325,343,362,381,401,421,442,463,485,507,530,553,577,601,626,651,677,703,730,757,785,813,842,
A033639 ,1,1,1,2,1,2,3,6,1,3,4,13,4,11,21,49,13,17,24,62,66,103,145,338,128,297,376,1156,763,1564,2592,6451,376,1532,1139,4235,4124,11714,8735,26105,5263,21212,18122,77153,35210,100649,135748,369972,95275,207638,
-A033640 ,1,1,2,1,3,7,6,20,52,6,26,104,32,162,460,356,1438,4048,712,3588,15272,5012,27460,90476,64944,300816,912472,90476,155420,611656,1067892,1770024,4763360,4151704,14746316,39566064,8915064,27813084,109938548,
+A033640 ,1,1,2,1,3,7,6,20,52,6,26,104,32,162,460,356,1438,4048,712,3588,15272,5012,27460,90476,64944,300816,912472,90476,155420,611656,1067892,1770024,4763360,4151704,14746316,39566064,8915064,27813084,109938548,76294212,222960908,
A033641 ,1,1,2,6,2,8,18,62,36,160,392,1496,1176,5664,14856,61560,5664,20520,102600,313464,123120,539184,1514952,5207160,3569088,15498360,43342056,164591976,145524528,682642512,1966450584,8092803816,1365285024,
A033642 ,1,1,2,6,24,6,30,66,228,978,456,2412,5736,22032,99600,66096,364896,928080,3878928,18299952,15515712,88715520,239493888,1073343744,5251350528,239493888,1312837632,7877025792,23870571264,96795122688,
A033643 ,1,1,2,6,24,120,24,144,312,1080,4632,24240,9264,57744,134016,517536,2338176,12725952,7014528,45192384,111428352,469862208,2213733888,12478256064,8854935552,58767959808,152955661824,693938824704,
A033644 ,1,1,2,6,24,120,720,120,840,1800,6240,26760,140040,867000,280080,2014080,4588320,17793120,80349120,437331840,2784689280,1311995520,9666063360,23268113280,98802529920,465014459520,2621479887360,
A033645 ,1,1,2,6,24,120,720,5040,720,5760,12240,42480,182160,953280,5901840,42266160,11803680,96336000,216279360,841510080,3798599040,20676015360,131653290240,962925062400,394959870720,3283735057920,7752349728000,
A033646 ,1,1,2,6,24,120,720,5040,40320,5040,45360,95760,332640,1426320,7464240,46211760,330946560,2693784240,661893120,6049461600,13422709440,52367051520,236313624960,1286302227840,8190440616960,59905688774400,
-A033647 ,1,1,2,6,24,120,720,5040,40320,362880,40320,403200,846720,2943360,12620160,66044160,408885120,2928240000,23834805120,217441486080,47669610240,482552582400,1060444385280,4146438320640,18706642053120,
+A033647 ,1,1,2,6,24,120,720,5040,40320,362880,40320,403200,846720,2943360,12620160,66044160,408885120,2928240000,23834805120,217441486080,47669610240,482552582400,1060444385280,4146438320640,18706642053120,101826086906880,648369805547520,
A033648 ,3,6,12,33,66,132,363,726,1353,4884,9768,18447,92928,175857,934428,1758867,9447438,17794887,96644658,182289327,906271608,1712444217,8836886388,17673772776,85401510447,159803020905,668823329856,1327746658722,3606313135953,7201626272016,
A033649 ,5,10,11,22,44,88,176,847,1595,7546,14003,44044,88088,176176,847847,1596595,7553546,14007103,44177144,88354288,176599676,853595347,1597190705,6668108656,13236127322,35608290553,
A033650 ,7,14,55,110,121,242,484,968,1837,9218,17347,91718,173437,907808,1716517,8872688,17735476,85189247,159487405,664272356,1317544822,3602001953,7193004016,13297007933,47267087164,
@@ -34697,7 +34697,7 @@ A034692 ,1,2,5,23,455,197447,38895873863,1512881323770591465287,2288809899755012
A034693 ,1,1,2,1,2,1,4,2,2,1,2,1,4,2,2,1,6,1,10,2,2,1,2,3,4,2,4,1,2,1,10,3,2,3,2,1,4,5,2,1,2,1,4,2,4,1,6,2,4,2,2,1,2,2,6,2,4,1,12,1,6,5,2,3,2,1,4,2,2,1,8,1,4,2,2,3,6,1,4,3,2,1,2,4,12,2,4,1,2,2,6,3,4,3,2,1,4,2,
A034694 ,2,3,7,5,11,7,29,17,19,11,23,13,53,29,31,17,103,19,191,41,43,23,47,73,101,53,109,29,59,31,311,97,67,103,71,37,149,191,79,41,83,43,173,89,181,47,283,97,197,101,103,53,107,109,331,113,229,59,709,61,367,311,
A034695 ,1,6,6,21,6,36,6,56,21,36,6,126,6,36,36,126,6,126,6,126,36,36,6,336,21,36,56,126,6,216,6,252,36,36,36,441,6,36,36,336,6,216,6,126,126,36,6,756,21,126,36,126,6,336,36,336,36,36,6,756,6,36,126,462,36,216,6,126,
-A034696 ,4,12,20,37,44,82,68,118,117,182,124,296,164,274,298,375,236,512,268,612,462,502,332,950,509,650,642,924,436,1310,508,1108,858,910,970,1831,628,1054,1078,1942,716,2034,
+A034696 ,4,12,20,37,44,82,68,118,117,182,124,296,164,274,298,375,236,512,268,612,462,502,332,950,509,650,642,924,436,1310,508,1108,858,910,970,1831,628,1054,1078,1942,716,2034,764,1680,1764,1294,844,2968,1197,2136,1522,
A034697 ,1,1,2,3,2,3,4,5,2,3,4,5,4,5,6,5,2,3,4,5,4,7,6,7,4,3,6,3,6,7,6,7,2,7,4,7,4,5,6,7,4,5,8,9,6,5,8,9,4,5,4,5,6,7,4,7,6,7,8,9,6,7,8,7,2,7,8,9,4,9,8,9,4,5,6,5,6,9,8,9,
A034698 ,2,7,31,113,233,647,1487,4919,6329,7951,26833,47737,53623,128959,135697,142327,1312777,3178289,6061607,26564393,32426081,102958417,209074609,323901311,587709359,1006009759,1029482303,9876033449,11061524183,15821898167,27926616007,
A034699 ,1,2,3,4,5,3,7,8,9,5,11,4,13,7,5,16,17,9,19,5,7,11,23,8,25,13,27,7,29,5,31,32,11,17,7,9,37,19,13,8,41,7,43,11,9,23,47,16,49,25,17,13,53,27,11,8,19,29,59,5,61,31,9,64,13,11,67,17,23,7,71,9,73,37,25,19,11,13,79,
@@ -41003,9 +41003,9 @@ A040998 ,2,5,11,29,41,59,71,89,101,131,149,179,181,191,199,211,239,241,251,269,2
A040999 ,3,7,13,17,19,23,31,37,43,47,53,61,67,73,79,83,97,103,107,109,113,127,137,139,151,157,163,167,173,193,197,223,227,229,233,257,263,271,277,283,293,307,313,317,331,337,347,349,353,367,373,383,397,
A041000 ,2,3,5,7,0,2,6,8,1,7,2,4,3,1,3,2,4,5,1,6,4,2,5,1,2,0,2,6,8,1,4,1,3,5,4,3,1,2,0,3,1,6,7,5,1,1,0,1,3,5,2,4,1,2,0,1,1,4,2,5,3,4,4,1,1,3,1,1,0,2,1,1,2,1,1,2,2,1,3,5,4,1,0,2,2,3,1,2,1,1,3,2,3,4,4,2,4,2,0,0,2,3,2,2,1,5,
A041001 ,1,14,125,906,5810,34364,191901,1026610,5312230,26767940,131990066,639210404,3048892740,14354652152,66828135005,308078809794,1408022619806,6385966846580,28765327498278,128777533131500,
-A041002 ,1,3,4,7,14,18,19,20,22,23,25,26,28,29,30,31,35,36,37,38,40,41,42,45,48,49,50,52,
-A041003 ,1,1,1,2,3,3,4,6,7,8,10,12,14,16,20,23,26,30,36,41,47,55,64,73,83,96,111,125,144,165,187,211,241,272,306,346,391,439,493,553,622,695,779,871,974,1086,1211,1348,1502,1671,
-A041004 ,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,1,0,0,0,1,1,1,1,0,1,1,1,0,0,1,0,0,1,1,1,0,1,
+A041002 ,1,3,4,7,14,18,19,20,22,23,25,26,28,29,30,31,35,36,37,38,40,41,42,45,48,49,50,52,54,55,57,59,63,67,70,71,73,79,80,83,85,87,90,91,93,94,98,100,101,102,103,106,108,110,111,112,116,117,121,124,132,135,137,142,143,144,
+A041003 ,1,1,1,2,3,3,4,6,7,8,10,12,14,16,20,23,26,30,36,41,47,55,64,73,83,96,111,125,144,165,187,211,241,272,306,346,391,439,493,553,622,695,779,871,974,1086,1211,1348,1502,1671,1857,2061,2288,2533,2808,3107,3439,3800,4199,4634,5113,
+A041004 ,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,1,0,0,0,1,1,1,1,0,1,1,1,0,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,1,
A041005 ,1,16,159,1260,8722,55152,326811,1844084,10015566,52754624,270976342,1362986520,6734927460,32775704608,157408497171,747269225028,3511471892470,16351481223840,75525932249922,346305571781224,
A041006 ,2,5,22,49,218,485,2158,4801,21362,47525,211462,470449,2093258,4656965,20721118,46099201,205117922,456335045,2030458102,4517251249,20099463098,44716177445,198964172878,442644523201,1969542265682,4381729054565,19496458483942,
A041007 ,1,2,9,20,89,198,881,1960,8721,19402,86329,192060,854569,1901198,8459361,18819920,83739041,186298002,828931049,1844160100,8205571449,18255302998,81226783441,180708869880,
@@ -46151,7 +46151,7 @@ A046146 ,0,0,1,2,3,3,5,5,0,5,7,8,0,11,5,0,0,14,11,15,0,0,19,21,0,23,19,23,0,27,0
A046147 ,1,2,3,2,3,5,3,5,2,5,3,7,2,6,7,8,2,6,7,11,3,5,3,5,6,7,10,11,12,14,5,11,2,3,10,13,14,15,7,13,17,19,5,7,10,11,14,15,17,19,20,21,2,3,8,12,13,17,22,23,7,11,15,19,2,5,11,14,20,23,2,3,8,10,11,14,15,18,19,21,26,
A046148 ,10,1,9,12,20,2430,5229,7448,282852,88200,8015040,200676960,2701775518,24655323238,15765750,1715313600,59049874884,1112489914536,14162129381400,135917876094000,1050596838951660,6832549561749912,38554260751029408,193081920969057120,
A046149 ,0,77,679,6788,68889,168889,2677889,26888999,126888999,3778888999,13778888999,113778888999,1113778888999,11113778888999,277777788888899,1277777788888899,11277777788888899,111277777788888899,
-A046150 ,9,77,976,8876,98886,997762,9999996,99988862,999888621,9998888773,99988887731,999888877311,9998888773111,99988887731111,998888887777772,9988888877777721,99999999998777772,999999999987777721,
+A046150 ,9,77,976,8876,98886,997762,9999996,99988862,999888621,9998888773,99988887731,999888877311,9998888773111,99988887731111,998888887777772,9988888877777721,99999999998777772,999999999987777721,9999999999877777211,99999999998777772111,
A046151 ,0,1,5,7,19,11,41,31,53,39,109,47,155,83,119,127,271,107,341,159,251,219,505,191,499,311,485,335,811,239,929,511,659,543,839,431,1331,683,935,639,1639,503,1805,879,1079,1011,2161,767,2057,999,
A046152 ,2,3,7,9,21,13,43,33,55,41,111,49,157,85,121,129,273,109,343,161,253,221,507,193,501,313,487,337,813,241,931,513,661,545,841,433,1333,685,937,641,1641,505,1807,881,1081,1013,2163,769,2059,1001,
A046153 ,3,11,79,23,127,191,1114111,1151,5119,6143,654311423,172031,88774955854727217151,1618481116086271,107221699928436768767,7421703487487,120946279055359,145135534866431,47287796087390207,
@@ -46165,11 +46165,11 @@ A046160 ,2,5,6,9,11,14,15,18,21,23,27,30,33,
A046161 ,1,2,8,16,128,256,1024,2048,32768,65536,262144,524288,4194304,8388608,33554432,67108864,2147483648,4294967296,17179869184,34359738368,274877906944,549755813888,2199023255552,4398046511104,70368744177664,
A046162 ,0,1,4,3,16,25,12,49,64,27,100,121,48,169,196,75,256,289,108,361,400,147,484,529,192,625,676,243,784,841,300,961,1024,363,1156,1225,432,1369,1444,507,1600,1681,588,1849,1936,675,2116,2209,768,2401,2500,
A046163 ,1,7,13,7,31,43,19,73,91,37,133,157,61,211,241,91,307,343,127,421,463,169,553,601,217,703,757,271,871,931,331,1057,1123,397,1261,1333,469,1483,1561,547,1723,1807,631,1981,2071,721,2257,2353,817,2551,2653,
-A046164 ,0,10,112,512,
+A046164 ,0,10,112,512,2138,7676,26034,87388,283436,910035,
A046165 ,1,1,2,8,49,462,6424,129425,3731508,152424420,8780782707,710389021036,80610570275140,12815915627480695,2855758994821922882,892194474524889501292,391202163933291014701953,240943718535427829240708786,208683398342300491409959279244,
A046166 ,0,0,0,0,1,171,17066,1298346,83384427,4762648737,249485204452,12226539786912,568267449522773,25296121946918823,1086375882592194558,45264846407024660598,1837809636559394481439,72965749033508656346829,
A046167 ,0,0,0,0,0,1,420,100814,18151560,2723868147,359750257020,43199991728608,4817721638970240,506352103838393813,50691406225311551700,4872449650707855334482,452435489306282260691640,
-A046168 ,0,0,0,0,0,0,0,1,2259,2835075,2609269245,1964842113102,1282310235724518,751046094737039530,403981152738311915910,202813559477327441603103,96164301413883629093787477,
+A046168 ,0,0,0,0,0,0,0,1,2259,2835075,2609269245,1964842113102,1282310235724518,751046094737039530,403981152738311915910,202813559477327441603103,96164301413883629093787477,43451168714121294538702462965,18840049876438047870808567312395,
A046169 ,0,0,0,0,0,0,0,0,1,5065,14109865,28586753635,47057782540912,66738127617591430,84508389361603849070,97838285747685771503930,105306724888534860425617883,106678207181565103216667658695,
A046170 ,1,2,5,12,30,73,183,456,1151,2900,7361,18684,47652,121584,311259,797311,2047384,5260692,13542718,34884239,89991344,232282110,600281932,1552096361,4017128206,10401997092,26957667445,69892976538,
A046171 ,1,2,5,13,36,98,272,740,2034,5513,15037,40617,110188,296806,802075,2155667,5808335,15582342,41889578,112212146,301100754,805570061,2158326727,5768299665,15435169364,41214098278,110164686454,
@@ -46217,18 +46217,18 @@ A046212 ,1,1,1,6,1,30,1,140,1,630,1,2772,1,12012,1,51480,1,218790,1,923780,1,387
A046213 ,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,2,3,2,1,1,1,1,5,2,3,1,5,2,1,1,1,1,7,2,11,2,11,2,7,2,1,1,1,1,9,2,9,1,11,1,9,1,9,2,1,1,1,1,11,2,27,2,20,1,20,1,27,2,11,2,1,1,1,1,13,2,19,1,67,2,40,1,67,2,19,1,13,2,1,1,1,1,15,2,
A046214 ,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,3,2,3,1,1,1,1,2,5,1,3,2,5,1,1,1,1,2,7,2,11,2,11,2,7,1,1,1,1,2,9,1,9,1,11,1,9,2,9,1,1,1,1,2,11,2,27,1,20,1,20,2,27,2,11,1,1,1,1,2,13,1,19,2,67,1,40,2,67,1,19,2,13,1,1,1,1,2,15,2,
A046215 ,2,3,2,3,2,5,2,3,5,2,7,2,11,2,11,2,7,2,9,2,9,11,9,9,2,11,2,27,2,20,20,27,2,11,2,13,2,19,67,2,40,67,2,19,13,2,15,2,51,2,105,2,147,2,147,2,105,2,51,2,15,2,17,2,33,78,126,147,126,78,33,17,2,19,2,83,2,111,204,
-A046216 ,2,2,3,2,3,2,5,3,2,5,2,7,2,11,2,11,2,7,2,9,9,11,9,2,9,2,11,2,27,20,20,2,27,2,11,2,13,19,2,67,40,2,67,19,2,13,2,15,2,51,
+A046216 ,2,2,3,2,3,2,5,3,2,5,2,7,2,11,2,11,2,7,2,9,9,11,9,2,9,2,11,2,27,20,20,2,27,2,11,2,13,19,2,67,40,2,67,19,2,13,2,15,2,51,2,105,2,147,2,147,2,105,2,51,2,15,2,17,33,78,126,147,126,78,33,2,17,2,
A046217 ,3,3,5,3,5,7,11,11,7,9,9,11,9,9,11,27,20,20,27,11,13,19,67,40,67,19,13,15,51,105,147,147,105,51,15,17,33,78,126,147,126,78,33,17,19,83,111,204,273,273,204,111,83,19,21,51,305,315,477,546,477,315,305,51,21,23,
-A046218 ,1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,20,20,27,11,1,1,13,19,67,40,67,19,13,1,1,15,51,105,147,147,105,51,
-A046219 ,1,1,1,1,2,1,1,2,2,1,1,2,1,2,1,1,2,2,2,2,1,1,2,1,1,1,2,1,1,2,2,1,1,2,2,1,1,2,1,2,1,2,1,2,1,1,2,2,2,2,2,2,2,2,1,1,2,
+A046218 ,1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,20,20,27,11,1,1,13,19,67,40,67,19,13,1,1,15,51,105,147,147,105,51,15,1,1,17,33,78,126,147,126,78,33,17,1,1,19,83,111,204,
+A046219 ,1,1,1,1,2,1,1,2,2,1,1,2,1,2,1,1,2,2,2,2,1,1,2,1,1,1,2,1,1,2,2,1,1,2,2,1,1,2,1,2,1,2,1,2,1,1,2,2,2,2,2,2,2,2,1,1,2,1,1,1,1,1,1,1,2,1,1,2,2,1,1,1,1,1,1,2,2,1,1,2,1,2,1,1,1,1,1,
A046220 ,1,2,3,5,7,11,9,27,20,13,19,67,40,15,51,105,147,17,33,78,126,83,111,204,273,21,305,315,477,546,23,123,407,935,792,1023,25,73,265,671,2519,1815,2046,171,338,936,3861,6149,29,99,847,1274,5733,5005,13871,7722,
-A046221 ,1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,27,11,1,1,13,19,67,67,19,13,1,1,15,51,105,147,147,105,51,
+A046221 ,1,1,1,1,1,1,1,3,3,1,1,5,3,5,1,1,7,11,11,7,1,1,9,9,11,9,9,1,1,11,27,27,11,1,1,13,19,67,67,19,13,1,1,15,51,105,147,147,105,51,15,1,1,17,33,147,33,17,1,1,19,83,111,273,273,111,83,19,1,1,21,51,
A046222 ,1,1,1,2,3,1,11,1,40,1,147,1,546,1,2046,1,7722,1,29315,1,111826,1,428298,1,1646008,1,6344366,1,24515700,1,94942620,1,368404110,1,1431985635,1,5574725970,1,21732560850,1,84828633120,1,331488081210,1,
A046223 ,1,1,2,1,1,3,1,11,1,40,1,147,1,546,1,2046,1,7722,1,29315,1,111826,1,428298,1,1646008,1,6344366,1,24515700,1,94942620,1,368404110,1,1431985635,1,5574725970,1,21732560850,1,84828633120,1,331488081210,1,
A046224 ,1,2,3,11,40,147,546,2046,7722,29315,111826,428298,1646008,6344366,24515700,94942620,368404110,1431985635,5574725970,21732560850,84828633120,331488081210,1296712152060,5077282282020,19897457591700,78039200913102,306302623291476,
-A046225 ,1,1,1,1,3,2,1,1,5,2,1,1,11,2,7,2,1,1,9,1,9,2,1,1,20,1,27,2,11,2,1,1,67,2,19,1,13,2,1,1,147,2,105,2,51,2,15,2,1,1,126,1,78,1,33,1,17,2,1,1,
-A046226 ,1,1,1,1,2,3,1,1,2,5,1,1,2,11,2,7,1,1,1,9,2,9,1,1,1,20,2,27,2,11,1,1,2,67,1,19,2,13,1,1,2,147,2,105,2,51,2,15,1,1,1,126,1,78,1,33,2,17,1,1,
-A046227 ,3,2,5,2,11,2,7,2,9,9,2,20,27,2,11,2,67,2,19,13,2,147,2,105,2,51,2,15,2,126,78,33,17,2,
+A046225 ,1,1,1,1,3,2,1,1,5,2,1,1,11,2,7,2,1,1,9,1,9,2,1,1,20,1,27,2,11,2,1,1,67,2,19,1,13,2,1,1,147,2,105,2,51,2,15,2,1,1,126,1,78,1,33,1,17,2,1,1,273,1,204,1,111,1,83,2,19,2,1,1,477,1,315,1,
+A046226 ,1,1,1,1,2,3,1,1,2,5,1,1,2,11,2,7,1,1,1,9,2,9,1,1,1,20,2,27,2,11,1,1,2,67,1,19,2,13,1,1,2,147,2,105,2,51,2,15,1,1,1,126,1,78,1,33,2,17,1,1,1,273,1,204,1,111,2,83,2,19,1,1,1,477,1,315,2,
+A046227 ,3,2,5,2,11,2,7,2,9,9,2,20,27,2,11,2,67,2,19,13,2,147,2,105,2,51,2,15,2,126,78,33,17,2,273,204,111,83,2,19,2,477,315,305,2,51,21,2,1023,792,935,2,407,2,123,2,23,2,1815,2519,2,671,265,73,25,2,3861,
A046228 ,3,5,11,7,9,9,20,27,11,67,19,13,147,105,51,15,126,78,33,17,273,204,111,83,19,477,315,305,51,21,1023,792,935,407,123,23,1815,2519,671,265,73,25,3861,6149,3861,936,338,171,27,13871,5005,5733,1274,847,99,29,29315,
A046229 ,1,1,1,1,3,1,1,5,1,1,11,7,1,1,9,1,9,1,1,20,1,27,11,1,1,67,19,1,13,1,1,147,105,51,15,1,1,126,1,78,1,33,1,17,1,1,273,1,204,1,111,1,83,19,1,1,477,1,315,1,305,51,1,21,1,1,1023,1,792,1,935,407,123,23,1,1,1815,1,
A046230 ,1,1,1,1,3,1,1,5,1,1,11,7,1,1,1,9,9,1,1,1,20,27,11,1,1,67,1,19,13,1,1,147,105,51,15,1,1,1,126,1,78,1,33,17,1,1,1,273,1,204,1,111,83,19,1,1,1,477,1,315,305,1,51,21,1,1,1,1023,1,792,935,407,123,23,1,1,1,1815,
@@ -46347,7 +46347,7 @@ A046342 ,1,1,3,8,24,74,245,815,2796,9707,34186,121562,436298,1577310,5740299,210
A046343 ,4,5,6,6,7,7,9,8,8,8,9,10,13,9,10,15,9,11,10,10,14,19,12,10,21,16,11,12,15,11,25,11,14,12,20,17,11,16,13,22,31,12,33,13,12,18,16,21,26,14,12,39,13,23,18,18,13,12,43,14,22,45,32,17,13,20,27,34,49,24,13,16,17,
A046344 ,6,8,10,10,9,14,12,16,11,14,20,16,22,13,18,26,13,18,12,22,32,20,34,24,17,15,40,28,19,24,22,44,15,46,26,14,50,24,34,17,23,36,56,30,19,26,25,17,62,64,42,28,16,21,70,36,46,29,30,74,48,38,76,30,16,21,52,82,15,19,
A046345 ,4,5,6,6,13,14,15,16,16,18,17,17,40,22,50,30,25,103,57,42,35,24,17,133,25,52,77,104,36,43,21,25,134,105,31,59,40,44,229,37,84,26,34,106,108,20,112,114,45,118,33,24,29,106,24,315,60,38,49,45,30,23,38,108,242,78,
-A046346 ,4,16,27,30,60,70,72,84,105,150,180,220,231,240,256,286,288,308,378,440,450,476,528,540,560,576,588,594,624,627,646,648,650,728,800,805,840,884,897,900,945,960,1008,1040,1056,1080,1100,1122,1134,1160,1170,
+A046346 ,4,16,27,30,60,70,72,84,105,150,180,220,231,240,256,286,288,308,378,440,450,476,528,540,560,576,588,594,624,627,646,648,650,728,800,805,840,884,897,900,945,960,1008,1040,1056,1080,1100,1122,1134,1160,1170,1248,
A046347 ,27,105,231,627,805,897,945,1581,1755,2079,2625,2967,3055,3125,3861,4185,4543,5355,5445,5487,5967,6075,6461,6525,6745,7881,8085,8505,8883,9555,9717,10125,10647,10707,11375,11385,12675,12789,13005,13275,13475,
A046348 ,4,646,2772,5445,8778,30303,48384,50505,54145,63336,77077,117711,219912,234432,239932,255552,272272,294492,535535,585585,636636,717717,825528,888888,951159,999999,1103011,1112111,1345431,2248422,2267622,
A046349 ,4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,27,28,30,32,33,35,36,40,42,44,45,48,49,50,54,55,56,60,63,64,66,70,72,75,77,80,81,84,88,90,96,98,99,100,105,108,110,112,120,121,125,126,128,132,135,140,144,147,150,
@@ -46396,7 +46396,7 @@ A046391 ,15015,19635,21945,23205,25935,26565,31395,33495,33915,35805,36465,39585
A046392 ,6,22,33,55,77,111,141,161,202,262,303,323,393,454,505,515,535,545,565,626,707,717,737,767,818,838,878,898,939,949,959,979,989,1111,1441,1661,1991,3113,3223,3443,3883,7117,7447,7997,9119,9229,9449,10001,
A046393 ,66,222,282,434,474,494,555,595,606,646,777,969,1001,1221,1551,1771,2222,2882,3333,3553,4334,4994,5335,5555,5665,5885,5995,6226,6446,6886,7337,7557,7667,7777,7887,8338,8558,8998,9339,9669,9779,9889,11211,
A046394 ,858,2002,2442,3003,4774,5005,5115,6666,10101,15351,17871,22422,22722,24242,26562,26962,28482,35853,36363,41314,43734,43834,45654,47874,49494,49794,49894,51015,51315,51415,53535,53835,53935,56865,58485,
-A046395 ,6006,8778,20202,28182,40404,41514,43134,50505,60606,63336,66066,68586,80808,83538,86268,87978,111111,141141,168861,171171,202202,204402,209902,210012,212212,219912,225522,231132,232232,239932,246642,249942,252252,258852,
+A046395 ,6006,8778,20202,28182,41514,43134,50505,68586,87978,111111,141141,168861,202202,204402,209902,246642,249942,262262,266662,303303,323323,393393,399993,438834,454454,505505,507705,515515,516615,519915,534435,535535,543345,
A046396 ,222222,282282,474474,555555,606606,646646,969969,2040402,2065602,2206022,2417142,2646462,2673762,2875782,3262623,3309033,4179714,4192914,4356534,4585854,4912194,5021205,5169615,5174715,5578755,
A046397 ,22444422,24266242,26588562,35888853,36399363,43777734,47199174,51066015,53588535,53888835,55233255,59911995,60066006,62588526,62700726,62888826,81699618,87788778,89433498,122434221,202040202,
A046398 ,244868442,1346776431,2012112102,2050550502,2222442222,2274994722,2402442042,2435775342,2601661062,2615775162,2806886082,4116996114,4163773614,4188998814,4305335034,4501551054,4515665154,4542992454,
@@ -46461,15 +46461,15 @@ A046456 ,128,2187,78125,496875,823543,1908795,3680733,6280989,11447205,11639595,
A046457 ,256,6561,390625,2484375,5764801,13361565,25765131,69090879,89994375,110875779,125919255,128035545,188245551,207847185,210815759,214358881,242810337,246891183,246944565,272828031,400792119,464794365,475376391,
A046458 ,4,6,8,12,14,20,32,38,
A046459 ,0,1,8,17,18,26,27,
-A046460 ,0,3,2,2,3,8,2,5,4,3,6,6,3,3,3,4,5,6,6,10,8,6,4,6,5,9,8,7,4,7,3,6,6,2,8,9,4,4,6,9,5,7,4,7,10,7,5,8,6,10,3,9,8,14,5,5,6,4,4,8,3,8,5,10,
+A046460 ,0,3,2,2,3,8,2,5,4,3,6,6,3,3,3,4,5,6,6,10,8,6,4,6,5,9,8,7,4,7,3,6,6,2,8,9,4,4,6,9,5,7,4,7,10,7,5,8,6,10,3,9,8,14,5,5,6,4,4,8,3,8,5,10,6,8,3,10,7,6,6,7,3,9,6,6,5,6,6,10,6,7,6,7,6,
A046461 ,3,4,7,34,97,
-A046462 ,2,5,10,13,14,15,31,51,61,
+A046462 ,2,5,10,13,14,15,31,51,61,67,73,
A046463 ,9,16,23,29,37,38,43,58,59,
-A046464 ,8,17,25,41,47,55,56,63,
-A046465 ,11,12,18,19,22,24,32,33,39,49,57,
-A046466 ,28,30,42,44,46,
-A046467 ,6,21,27,35,48,53,60,62,
-A046468 ,26,36,40,52,
+A046464 ,8,17,25,41,47,55,56,63,77,94,101,103,107,
+A046465 ,11,12,18,19,22,24,32,33,39,49,57,65,70,71,75,76,78,79,81,83,85,87,88,91,95,96,99,105,110,
+A046466 ,28,30,42,44,46,69,72,82,84,86,93,109,116,
+A046467 ,6,21,27,35,48,53,60,62,66,92,106,108,117,
+A046468 ,26,36,40,52,74,90,102,112,114,115,
A046469 ,0,1,2,3,4,5,6,7,8,9,343,736,1285,2187,2592,3125,3685,3972,4096,14641,15552,15632,15642,15645,15655,15656,15662,15667,15698,16384,17536,23328,32771,32785,37179,39369,39372,39382,43775,45927,45947,46660,
A046470 ,2,8,12,18,20,28,30,32,42,44,48,50,52,66,68,70,72,76,78,80,92,98,102,108,110,112,114,116,120,124,128,130,138,148,154,162,164,168,170,172,174,176,180,182,186,188,190,192,200,208,212,222,230,236,238,242,244,
A046471 ,8,1,5,5,4,4,8,3,3,6,3,1,11,5,7,6,4,2,9,3,3,7,3,3,13,4,2,6,5,1,10,1,7,3,5,2,8,2,2,6,1,4,9,5,3,8,8,4,11,1,3,4,4,5,2,1,6,3,4,4,5,2,3,4,4,3,8,1,5,3,2,2,5,4,5,3,3,4,8,4,2,4,4,1,5,2,6,6,3,2,7,3,3,8,5,1,7,1,4,5,2,3,9,
@@ -46481,7 +46481,7 @@ A046476 ,2,3,5,16561,9470749,90750705709,
A046477 ,2,3,5,7,373,13331,30103,1496941,1970791,
A046478 ,2,3,5,7,191,373,3106013,1400232320041,
A046479 ,2,3,5,7,787,38183,3286823,998111899,999454999,
-A046480 ,2,3,7,11,181,797,713171317,
+A046480 ,2,3,5,7,11,181,797,713171317,
A046481 ,2,3,5,7,11,313,353,797,
A046482 ,2,3,5,7,11,9046409,14203330241,
A046483 ,2,3,5,7,11,919,
@@ -46532,42 +46532,42 @@ A046527 ,1,1,1,2,5,1,5,22,9,1,14,93,58,13,1,42,386,325,110,17,1,132,1586,1686,76
A046528 ,1,3,7,21,31,93,127,217,381,651,889,2667,3937,8191,11811,24573,27559,57337,82677,131071,172011,253921,393213,524287,761763,917497,1040257,1572861,1777447,2752491,3120771,3670009,4063201,5332341,7281799,11010027,12189603,
A046529 ,1,9,145,1459,13139,213040,1758629,19210003,153362052,1724515947,15434111703,176062673167,1281243062759,11270705761846,142415666495594,1145004602355098,11968345165915849,121510372538044487,
A046530 ,1,2,3,3,5,6,3,5,3,10,11,9,5,6,15,10,17,6,7,15,9,22,23,15,21,10,7,9,29,30,11,19,33,34,15,9,13,14,15,25,41,18,15,33,15,46,47,30,15,42,51,15,53,14,55,15,21,58,59,45,21,22,9,37,25,66,23,51,69,30,71,15,25,26,63,
-A046531 ,1,1,3,1,5,1,11,7,1,9,9,1,20,27,11,1,67,19,13,1,147,105,51,15,1,126,78,33,17,1,
+A046531 ,1,1,3,1,5,1,11,7,1,9,9,1,20,27,11,1,67,19,13,1,147,105,51,15,1,126,78,33,17,1,273,204,111,83,19,1,477,315,305,51,21,1,1023,792,935,407,123,23,1,1815,2519,671,265,73,25,1,3861,6149,3861,936,338,171,
A046532 ,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,1,1,1,5,3,1,5,1,1,1,1,7,11,11,7,1,1,1,1,9,9,1,11,1,9,1,9,1,1,1,1,11,27,20,1,20,1,27,11,1,1,1,1,13,19,1,67,40,1,67,19,1,13,1,1,1,1,15,51,
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A046567 ,5,5,9,2,5,9,13,19,19,13,17,8,2,19,8,17,21,49,2,35,2,35,49,21,25,2,35,119,35,119,2,35,25,29,95,189,259,259,189,95,29,33,31,71,112,2,259,112,71,31,33,37,157,102,183,2,483,2,483,183,102,157,37,41,2,97,565,285,2,
A046568 ,1,1,1,1,1,1,1,5,5,1,1,9,5,9,1,1,13,19,19,13,1,1,17,8,19,8,17,1,1,21,49,35,35,49,21,1,1,25,35,119,35,119,35,25,1,1,29,95,189,259,259,189,95,29,1,1,33,31,71,112,259,112,71,31,33,1,
A046569 ,1,1,1,1,4,1,1,4,4,1,1,4,2,4,1,1,4,4,4,4,1,1,4,1,2,1,4,1,1,4,4,2,2,4,4,1,1,4,2,4,1,4,2,4,1,1,4,4,4,4,4,4,4,4,1,1,4,1,1,1,2,1,1,1,4,1,1,4,4,1,1,2,2,1,1,4,4,1,1,4,2,4,1,2,1,2,1,4,2,4,1,1,4,4,4,4,2,2,2,2,4,4,4,4,1,
@@ -46581,7 +46581,7 @@ A046576 ,1,1,1,1,4,5,1,1,4,9,1,1,4,19,4,13,1,1,1,8,4,17,1,1,2,35,4,49,4,21,1,1,4
A046577 ,4,5,4,9,4,19,4,13,8,4,17,2,35,4,49,4,21,4,119,2,35,4,25,4,259,4,189,4,95,4,29,112,71,31,4,33,2,483,183,102,4,157,4,37,2,849,285,4,565,2,97,4,41,2,1815,2,1419,4,1705,4,759,4,235,4,45,1617,4,4543,616,2,497,70,4,
A046578 ,5,9,19,13,8,17,2,35,49,21,119,2,35,25,259,189,95,29,112,71,31,33,2,483,183,102,157,37,2,849,285,565,2,97,41,2,1815,2,1419,1705,759,235,45,1617,4543,616,2,497,70,49,3432,11011,7007,2,1729,2,637,329,53,24739,2,9009,
A046579 ,1,1,1,1,5,1,1,9,1,1,19,13,1,1,8,1,17,1,1,35,2,49,21,1,1,119,35,2,25,1,1,259,189,95,29,1,1,112,1,71,1,31,1,33,1,1,483,2,183,1,102,1,157,37,1,1,849,2,285,1,565,97,2,41,1,1,1815,2,1419,2,1705,759,235,45,1,1,1617,
-A046580 ,1,1,5,1,9,1,19,13,1,8,17,1,35,49,21,1,119,35,25,1,259,189,95,29,1,112,71,31,33,1,483,189,102,157,37,1,861,291,565,97,41,1,1827,1443,1729,759,235,45,1,1635,4615,622,497,70,49,1,3462,11155,7103,1119,567,329,
+A046580 ,1,1,5,1,9,1,19,13,1,8,17,1,35,49,21,1,119,35,25,1,259,189,95,29,1,112,71,31,33,1,483,183,102,157,37,1,849,285,565,97,41,1,1815,1419,1705,759,235,45,1,1617,4543,616,497,70,49,1,3432,11011,7007,1729,637,
A046581 ,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5,1,1,1,1,9,5,2,9,1,1,1,1,13,19,19,13,1,1,1,1,17,8,1,19,2,8,1,17,1,1,1,1,21,49,35,2,35,2,49,21,1,1,1,1,25,35,2,119,35,1,119,35,2,25,1,1,1,1,29,
A046582 ,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5,1,1,1,1,9,2,5,9,1,1,1,1,13,19,19,13,1,1,1,1,17,1,8,2,19,1,8,17,1,1,1,1,21,49,2,35,2,35,49,21,1,1,1,1,25,2,35,119,1,35,119,2,35,25,1,1,1,1,29,
A046583 ,1,1,1,1,5,1,1,9,1,1,19,13,1,1,1,8,17,1,1,2,35,49,21,1,1,119,2,35,25,1,1,259,189,95,29,1,1,1,112,1,71,1,31,33,1,1,2,483,1,183,1,102,157,37,1,1,2,849,1,285,565,2,97,41,1,1,2,1815,2,1419,1705,759,235,45,1,1,1,
@@ -46595,7 +46595,7 @@ A046590 ,5,9,19,13,8,17,35,49,21,119,35,25,259,189,95,29,112,71,31,33,483,183,10
A046591 ,1,1,1,1,1,1,1,1,1,4,1,1,1,1,5,4,5,4,1,1,1,1,9,4,5,9,4,1,1,1,1,13,4,19,4,19,4,13,4,1,1,1,1,17,4,8,1,19,8,1,17,4,1,1,1,1,21,4,49,4,35,35,49,4,21,4,1,1,1,1,25,4,35,119,4,35,1,119,4,35,25,4,1,1,1,1,29,4,
A046592 ,1,1,1,1,1,1,1,1,4,1,1,1,1,1,4,5,4,5,1,1,1,1,4,9,5,4,9,1,1,1,1,4,13,4,19,4,19,4,13,1,1,1,1,4,17,1,8,19,1,8,4,17,1,1,1,1,4,21,4,49,35,35,4,49,4,21,1,1,1,1,4,25,35,4,119,1,35,4,119,35,4,25,1,1,1,1,4,29,
A046593 ,4,4,5,4,5,4,9,5,4,9,4,13,4,19,4,19,4,13,4,17,8,19,8,4,17,4,21,4,49,35,35,4,49,4,21,4,25,35,4,119,35,4,119,35,4,25,4,29,4,95,4,189,4,259,4,259,4,189,4,95,4,29,4,33,31,71,112,259,112,71,31,4,33,4,37,4,157,102,
-A046594 ,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5,1,1,1,1,9,5,9,1,1,1,1,13,19,19,13,1,1,1,1,17,1,8,19,1,8,17,1,1,1,1,21,49,35,35,49,21,1,1,1,1,25,35,119,1,35,119,35,25,1,1,1,1,29,
+A046594 ,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5,1,1,1,1,9,5,9,1,1,1,1,13,19,19,13,1,1,1,1,17,1,8,19,1,8,17,1,1,1,1,21,49,35,35,49,21,1,1,1,1,25,35,119,1,35,119,35,25,1,1,1,1,29,95,189,259,259,189,95,29,1,
A046595 ,4,4,4,4,2,4,4,4,4,4,4,2,4,4,4,2,2,4,4,4,2,4,4,2,4,4,4,4,4,4,4,4,4,4,2,4,4,4,2,2,4,4,4,2,4,2,2,4,2,4,4,4,4,4,2,2,2,2,4,4,4,4,4,2,4,4,2,4,4,4,2,2,4,4,4,4,2,2,4,4,4,2,4,4,2,4,4,2,4,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
A046596 ,1,1,1,1,4,1,1,4,4,1,1,4,4,1,1,4,4,4,4,1,1,4,1,1,4,1,1,4,4,4,4,1,1,4,4,1,4,4,1,1,4,4,4,4,4,4,4,4,1,1,4,1,1,1,1,1,1,4,1,1,4,4,1,1,1,1,4,4,1,1,4,4,1,1,1,4,4,1,1,4,4,4,4,4,4,4,4,1,1,4,1,1,4,1,1,1,4,1,1,4,1,1,4,4,4,
A046597 ,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,2,1,2,1,2,1,2,1,1,2,2,2,2,1,1,1,2,1,1,1,1,1,2,1,1,1,2,2,1,1,2,2,1,1,2,1,2,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,
@@ -46606,7 +46606,7 @@ A046601 ,1,1,1,1,1,1,1,1,1,5,1,1,1,1,6,5,6,5,1,1,1,1,11,5,12,5,11,5,1,1,1,1,16,5
A046602 ,1,1,1,1,1,1,1,1,5,1,1,1,1,1,5,6,5,6,1,1,1,1,5,11,5,12,5,11,1,1,1,1,5,16,5,23,5,23,5,16,1,1,1,1,5,21,5,39,5,46,5,39,5,21,1,1,1,1,5,26,1,12,1,17,1,17,1,12,5,26,1,1,1,1,5,31,5,86,1,29,1,34,1,29,5,86,5,31,1,1,1,
A046603 ,5,6,5,6,5,11,5,12,5,11,5,16,5,23,5,23,5,16,5,21,5,39,5,46,5,39,5,21,5,26,5,12,17,17,12,26,5,31,5,86,5,29,34,29,86,5,31,5,36,5,117,5,231,5,63,63,231,5,117,5,36,5,41,5,153,5,348,5,546,5,126,546,5,348,5,153,5,
A046604 ,5,5,6,5,6,5,11,5,12,5,11,5,16,5,23,5,23,5,16,5,21,5,39,5,46,5,39,5,21,5,26,12,17,17,12,5,26,5,31,5,86,29,34,29,5,86,5,31,5,36,5,117,5,231,63,63,5,231,5,117,5,36,5,41,5,153,5,348,5,546,126,5,546,5,348,5,153,5,
-A046605 ,6,6,11,12,11,16,23,23,16,21,39,46,39,21,26,12,17,17,12,26,31,86,29,34,29,86,31,36,117,231,63,63,231,117,36,41,153,348,546,126,546,348,153,41,46,199,501,894,1176,1176,894,501,199,46,51,49,140,279,414,2352,414,279,140,
+A046605 ,6,6,11,12,11,16,23,23,16,21,39,46,39,21,26,12,17,17,12,26,31,86,29,34,29,86,31,36,117,231,63,63,231,117,36,41,153,348,546,126,546,348,153,41,46,194,501,894,1176,1176,894,501,194,46,51,48,139,279,414,2352,414,279,139,
A046606 ,1,1,1,1,1,1,1,6,6,1,1,11,12,11,1,1,16,23,23,16,1,1,21,39,46,39,21,1,1,26,12,17,17,12,26,1,1,31,86,29,34,29,86,31,1,1,36,117,231,63,63,231,117,36,1,1,41,153,348,546,126,546,348,153,41,1,1,
A046607 ,1,1,1,1,5,1,1,5,5,1,1,5,5,5,1,1,5,5,5,5,1,1,5,5,5,5,5,1,1,5,1,1,1,1,5,1,1,5,5,1,1,1,5,5,1,1,5,5,5,1,1,5,5,5,1,1,5,5,5,5,1,5,5,5,5,1,1,5,5,5,5,5,5,5,5,5,5,1,1,5,1,1,1,1,5,1,1,1,1,5,1,1,5,5,1,1,1,5,5,1,1,1,5,5,1,
A046608 ,1,5,6,11,12,16,23,21,39,46,26,17,31,86,29,34,36,117,231,63,41,153,348,546,126,194,501,894,1176,51,48,139,279,414,2352,56,291,187,418,693,4422,61,347,1226,605,1111,7887,8844,66,408,1573,4251,1716,13442,16731,
@@ -46614,8 +46614,8 @@ A046609 ,1,1,1,1,1,1,1,1,1,11,11,1,1,23,23,1,1,21,39,39,21,1,1,17,17,1,1,31,29,2
A046610 ,1,1,1,5,12,5,46,5,34,1,126,1,2352,5,8844,5,33462,5,25454,1,97240,1,1864356,5,7171892,5,27665596,5,21395520,1,82907640,1,321868854,1,1251661518,1,4874644104,1,19010020260,1,74225053980,1,290134122564,1,
A046611 ,1,1,5,1,5,12,5,46,1,34,1,126,5,2352,5,8844,5,33462,1,25454,1,97240,5,1864356,5,7171892,5,27665596,1,21395520,1,82907640,1,321868854,1,1251661518,1,4874644104,1,19010020260,1,74225053980,1,290134122564,1,
A046612 ,1,5,12,46,34,126,2352,8844,33462,25454,97240,1864356,7171892,27665596,21395520,82907640,321868854,1251661518,4874644104,19010020260,74225053980,290134122564,1135234789728,4446052592904,17427428373420,
-A046613 ,1,1,1,1,6,5,1,1,11,5,1,1,23,5,16,5,1,1,39,5,21,5,1,1,17,1,12,1,26,5,1,1,29,1,86,5,31,5,1,1,63,1,231,5,117,5,36,5,1,1,546,5,348,5,153,5,41,5,1,1,
-A046614 ,1,1,1,1,5,6,1,1,5,11,1,1,5,23,5,16,1,1,5,39,5,21,1,1,1,17,1,12,5,26,1,1,1,29,5,86,5,31,1,1,1,63,5,231,5,117,5,36,1,1,5,546,5,348,5,153,5,41,1,1,
+A046613 ,1,1,1,1,6,5,1,1,11,5,1,1,23,5,16,5,1,1,39,5,21,5,1,1,17,1,12,1,26,5,1,1,29,1,86,5,31,5,1,1,63,1,231,5,117,5,36,5,1,1,546,5,348,5,153,5,41,5,1,1,1176,5,894,5,501,5,194,5,46,5,1,1,414,1,
+A046614 ,1,1,1,1,5,6,1,1,5,11,1,1,5,23,5,16,1,1,5,39,5,21,1,1,1,17,1,12,5,26,1,1,1,29,5,86,5,31,1,1,1,63,5,231,5,117,5,36,1,1,5,546,5,348,5,153,5,41,1,1,5,1176,5,894,5,501,5,194,5,46,1,1,1,414,1,
A046615 ,6,5,11,5,23,5,16,5,39,5,21,5,17,12,26,5,29,86,5,31,5,63,231,5,117,5,36,5,546,5,348,5,153,5,41,5,1176,5,894,5,501,5,194,5,46,5,414,279,139,48,51,5,4422,5,693,418,187,291,5,56,5,7887,5,1111,605,1226,5,347,5,61,
A046616 ,6,11,23,16,39,21,17,12,26,29,86,31,63,231,117,36,546,348,153,41,1176,894,501,194,46,414,279,139,48,51,4422,693,418,187,291,56,7887,1111,605,1226,347,61,16731,13442,1716,4251,1573,408,66,30173,22022,12831,
A046617 ,1,1,1,1,6,1,1,11,1,1,23,16,1,1,39,21,1,1,17,1,12,1,26,1,1,29,1,86,31,1,1,63,1,231,117,36,1,1,546,348,153,41,1,1,1176,894,501,194,46,1,1,414,1,279,1,139,1,48,1,51,1,1,4422,693,1,418,1,187,1,291,56,1,1,7887,
@@ -46698,7 +46698,7 @@ A046693 ,1,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10
A046694 ,1,667,252,601,684,171,531,178,372,168,469,123,629,385,309,388,611,55,672,630,449,491,92,632,57,106,88,580,173,185,366,666,27,538,429,379,622,456,269,136,87,280,36,632,160,556,435,345,194,14,570,52,209,652,172,542,49,
A046695 ,0,0,1,2,0,1,2,3,1,2,3,4,0,3,4,2,1,3,2,1,0,2,1,4,5,1,4,5,1,2,0,1,2,3,1,2,3,4,2,3,4,2,3,4,2,1,0,2,8,4,5,3,4,5,6,2,5,1,2,3,1,2,3,4,2,3,4,2,3,4,2,3,0,2,3,4,5,3,4,5,6,4,5,6,2,3,1,2,3,4,2,3,4,2,3,4,2,3,0,2,3,4,
A046696 ,1,2,5,13,21,31,47,73,99,125,151,177,315,409,
-A046697 ,1,1,3,9,35,137,574,2431,10534,46123,204343,912967,4111375,18637303,84988775,389589095,
+A046697 ,1,1,3,9,35,137,574,2431,10534,46123,204343,912967,4111375,18637303,84988775,389589095,1794280695,8298536715,38527147681,179487103411,838820394913,3931498654243,18475619837816,87036536948489,410947150379120,1944378509186237,
A046698 ,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
A046699 ,1,1,2,2,3,4,4,4,5,6,6,7,8,8,8,8,9,10,10,11,12,12,12,13,14,14,15,16,16,16,16,16,17,18,18,19,20,20,20,21,22,22,23,24,24,24,24,25,26,26,27,28,28,28,29,30,30,31,32,32,32,32,32,32,33,34,34,35,36,36,36,37,
A046700 ,1,1,1,2,3,4,4,5,5,7,6,8,7,9,8,10,9,12,9,12,10,15,11,16,12,17,13,18,14,19,15,21,17,18,18,21,19,21,24,20,22,24,20,31,19,30,19,28,19,30,21,32,21,34,21,36,21,38,21,48,18,53,15,49,18,49,24,61,17,51,19,42,
@@ -46884,7 +46884,7 @@ A046879 ,1,1,1,6,3,15,30,420,105,315,315,6930,3465,90090,180180,72072,9009,15315
A046880 ,1,0,0,0,0,1,1,1,2,5,7,9,24,19,35,46,86,134,187,259,450,616,823,1233,1799,2355,3342,4468,6063,8148,10774,13977,18769,23589,30683,39393,49878,62372,79362,98541,121354,151201,186611,225245,277930,335569,
A046881 ,5,65,1729,635318657,
A046882 ,1,1,4,46656,1333735776850284124449081472843776,
-A046883 ,17,99551,4303027,6440999,14968819,95517973,527737957,1893230839,1246492090901,12426836115943,21732382677641,154895576080181,
+A046883 ,17,99551,4303027,6440999,14968819,95517973,527737957,1893230839,1246492090901,12426836115943,21732382677641,154895576080181,2677628540590583,133475456543097857,820396537622790811,
A046884 ,17,99551,14968819,95517973,
A046885 ,1,4,18,85,411,2013,9933,49236,244750,1218888,6077644,30329434,151439158,756452890,3779590010,18888255205,94405918355,471899946985,2359022096225,11793343217935,58960151969255,294776293579255,
A046886 ,2,3,3,3,3,5,2,4,4,4,3,5,2,4,5,4,2,7,2,5,4,4,3,6,3,4,4,4,3,8,2,4,5,3,4,8,2,3,4,6,3,7,2,5,6,4,2,7,2,5,4,4,3,8,4,6,3,4,2,9,2,3,6,4,4,7,2,4,5,6,2,9,2,4,6,3,3,8,2,6,5,4,3,7,3,4,4,6,3,11,2,4,3,3,4,8,2,5,7,6,2,6,2,5,
@@ -47162,16 +47162,16 @@ A047157 ,1,2,3,24,104,316,1347,5007,19908,78909,300750,1172598,4544071,17726423,
A047158 ,1,3,7,36,54,449,1592,7243,30636,115153,466704,1833753,7311539,29170551,114837610,453949280,1790444065,7070681835,27947590455,110325729977,435614497712,1719775419793,6791219544205,26827316475982,
A047159 ,1,3,13,72,417,2509,15438,96310,607231,3859594,24689329,158756701,1025210362,6644336882,43193144567,281525730152,1839132807282,12038672624172,78943126865652,518485130526736,3410150902046028,
A047160 ,0,0,1,0,1,0,3,2,3,0,1,0,3,2,3,0,1,0,3,2,9,0,5,6,3,4,9,0,1,0,9,4,3,6,5,0,9,2,3,0,1,0,3,2,15,0,5,12,3,8,9,0,7,12,3,4,15,0,1,0,9,4,3,6,5,0,15,2,3,0,1,0,15,4,3,6,5,0,9,2,15,0,5,12,3,14,9,0,7,12,9,4,15,6,7,0,9,2,3,
-A047161 ,0,0,1,2,6,9,21,30,70,100,235,335,791,1127,2681,3822,9150,13050,31401,44802,108262,154517,374715,534963,1301235,1858155,4531423,6472167,15818791,22597759,55339849,79067374,193962894,277164294,
-A047162 ,0,0,0,2,3,9,12,30,40,100,135,335,455,1127,1540,3822,5250,13050,18000,44802,62007,154517,214467,534963,744315,1858155,2590679,6472167,9039823,22597759,31612324,79067374,110761494,277164294,
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-A047168 ,0,0,0,0,0,0,0,4,5,25,30,90,105,245,280,588,666,1458,1665,4125,4785,12705,14850,38830,45331,113399,131859,320411,371735,903175,1048840,2594540,3021240,7594920,
-A047169 ,0,0,0,0,0,0,0,0,4,5,30,36,126,147,392,448,1008,1134,2304,2565,5115,5687,12342,13860,34320,39039,104104,119119,315679,360815,913640,1039808,2508928,2842604,
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+A047170 ,0,0,0,0,0,0,0,0,0,5,6,36,42,147,168,448,504,1134,1260,2565,2827,5687,6270,13860,15444,39039,44044,119119,135135,360815,408968,1039808,1174224,2842604,3197904,7515054,8436114,19773414,22203020,53093125,59750649,146794494,
A047171 ,0,0,0,2,3,9,14,34,55,125,209,461,791,1715,3002,6434,11439,24309,43757,92377,167959,352715,646645,1352077,2496143,5200299,9657699,20058299,37442159,77558759,145422674,300540194,565722719,1166803109,2203961429,4537567649,
A047172 ,0,0,1,3,6,13,21,45,70,154,235,525,791,1793,2681,6153,9150,21206,31401,73359,108262,254607,374715,886171,1301235,3091971,4531423,10811671,15818791,37877401,55339849,132924649,193962894,467187454,
A047173 ,0,0,0,0,3,4,12,15,40,51,135,175,455,596,1540,2037,5250,7000,18000,24156,62007,83667,214467,290719,744315,1012935,2590679,3537703,9039823,12381252,31612324,43411489,110761494,152459010,
@@ -47938,15 +47938,15 @@ A047933 ,3,5,7,13,31,61,103,109,151,157,181,199,229,257,271,277,347,349,373,421,
A047934 ,2,3,5,11,29,59,101,107,149,151,179,197,227,251,269,271,337,347,367,419,461,659,733,821,827,971,991,1019,1021,1061,1091,1229,1277,1301,1427,1451,1619,1667,1787,1877,1931,1949,1997,2027,2141,2237,2267,2309,
A047935 ,1,2,2,2,2,2,2,2,2,6,2,2,2,6,2,6,10,2,6,2,2,2,6,2,2,6,6,2,10,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,6,6,2,6,2,6,6,2,6,2,2,2,2,2,2,2,2,2,2,2,6,2,2,10,2,2,2,2,6,2,6,2,2,2,2,6,2,2,2,2,10,6,10,2,2,2,10,2,2,2,6,10,
A047936 ,2,41,109,151,229,251,271,313,337,367,409,439,733,761,971,991,1021,1031,1069,1289,1297,1303,1429,1471,1489,1759,1783,1789,1811,1871,1873,1879,2137,2411,2441,2551,2749,2791,2971,3001,3061,3079,3109,3221,3229,
-A047937 ,1,2,6,140,16456,8390720,17179934976,140737496748032,4611686019501162496,604462909807864344215552,316912650057057631849169289216,664613997892457937028364283517337600,
-A047938 ,1,3,24,4995,10763361,211822552035,37523658921114744,59824832307866205347043,858420955073128584419531008641,110856622060759442496180656754310346403,
-A047939 ,1,4,70,65824,1073758336,281474993496064,1180591620734591303680,79228162514264619068554215424,85070591730234615870455337878516924416,1461501637330902918203686041642102636484130504704,
-A047940 ,1,5,165,489125,38147070625,74505806274453125,3637978807092666626953125,4440892098500626236200333251953125,135525271560688054250937420874834136962890625,
-A047941 ,1,6,336,2521476,705277897416,7107572010747738816,2578606199622659276537193216,33678386561031835867238388699701784576,15835071665743319426540573726368249140484891508736,
-A047942 ,1,7,616,10092775,8308234084801,335267154940213889575,662932711464913775048175499816,64230894380264719522488136461023341060807,
-A047943 ,1,8,1044,33562880,70368748374016,9444732965876730429440,81129638414606686199388699623424,44601490397061246283080881278262737180295168,
-A047944 ,1,9,1665,96870249,463255057977921,179474496923598616041129,5632099886234793640483695986653185,14316042242555870306568544190208626253583093449,
-A047945 ,1,10,2530,250025500,2500000025005000,2500000000002500005000000,250000000000000000250000000500000000,2500000000000000000000002500000000005000000000000,
+A047937 ,1,2,6,140,16456,8390720,17179934976,140737496748032,4611686019501162496,604462909807864344215552,316912650057057631849169289216,664613997892457937028364283517337600,5575186299632655785385110159782842147536896,187072209578355573530071668259090783437390809661440,
+A047938 ,1,3,24,4995,10763361,211822552035,37523658921114744,59824832307866205347043,858420955073128584419531008641,110856622060759442496180656754310346403,128844380183002832759115461915902241562318377784,1347757724935823407809884872163997148505019182125807296675,
+A047939 ,1,4,70,65824,1073758336,281474993496064,1180591620734591303680,79228162514264619068554215424,85070591730234615870455337878516924416,1461501637330902918203686041642102636484130504704,401734511064747568885490523085607563280607806359022338048000,
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+A047945 ,1,10,2530,250025500,2500000025005000,2500000000002500005000000,250000000000000000250000000500000000,2500000000000000000000002500000000005000000000000,2500000000000000000000000000000025000000000000005000000000000000,
A047946 ,3,2,8,17,48,122,323,842,2208,5777,15128,39602,103683,271442,710648,1860497,4870848,12752042,33385283,87403802,228826128,599074577,1568397608,4106118242,10749957123,28143753122,73681302248,
A047947 ,2,4,2,4,4,4,6,4,2,4,6,6,2,6,4,6,4,6,4,4,6,4,6,10,4,6,6,4,6,4,6,6,4,2,4,6,8,6,4,2,8,4,10,2,4,10,10,4,6,6,2,10,6,2,6,4,6,12,4,6,10,4,6,6,6,8,6,10,4,8,6,6,2,6,12,10,2,4,6,6,8,4,2,10,8,6,6,4,8,10,2,6,4,2,
A047948 ,47,151,167,251,257,367,557,587,601,647,727,941,971,1097,1117,1181,1217,1361,1741,1747,1901,2281,2411,2671,2897,2957,3301,3307,3631,3727,4007,4451,4591,4651,4987,5101,5107,5297,5381,5387,5557,5801,6067,6257,6311,
@@ -49337,7 +49337,7 @@ A049332 ,2,4,5,10,17,34,65,130,257,514,1025,2050,4097,8194,16385,32770,65537,131
A049333 ,1,2,243,4951760157141521099596496896,
A049334 ,1,0,1,0,0,1,1,0,0,0,2,2,1,1,0,0,0,0,3,5,5,4,2,1,0,0,0,0,0,6,13,19,22,19,13,5,2,0,0,0,0,0,0,11,33,67,107,130,130,96,51,16,5,0,0,0,0,0,0,0,23,89,236,486,804,1112,1211,1026,626,275,72,14,0,0,0,0,0,0,0,0,
A049335 ,1,240,2160,6720,17280,
-A049336 ,0,0,0,0,0,0,1,0,0,0,0,1,1,1,0,0,0,0,0,1,2,3,2,1,0,0,0,0,0,0,1,3,9,13,11,5,2,0,0,0,0,0,0,0,1,4,20,49,77,75,47,16,5,0,0,0,0,0,0,0,0,1,6,40,158,406,662,737,538,259,72,14,0,0,0,0,0,0,0,0,0,1,7,70,426,1645,4176,
+A049336 ,0,0,0,0,0,0,1,0,0,0,0,1,1,1,0,0,0,0,0,1,2,3,2,1,0,0,0,0,0,0,1,3,9,13,11,5,2,0,0,0,0,0,0,0,1,4,20,49,77,75,47,16,5,0,0,0,0,0,0,0,0,1,6,40,158,406,662,737,538,259,72,14,0,0,0,0,0,0,0,0,0,1,7,70,426,1645,4176,7307,8871,7541,4353,1671,378,50,
A049337 ,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,2,2,2,0,0,0,0,0,0,0,0,0,0,0,2,8,11,8,5,0,0,0,0,0,0,0,0,0,0,0,0,2,11,42,74,76,38,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,74,296,633,768,558,219,
A049338 ,1,1,1,2,5,14,50,233,1249,7616,
A049339 ,1,1,2,3,6,14,36,121,533,3067,21798,
@@ -51043,8 +51043,8 @@ A051038 ,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,20,21,22,24,25,27,28,30,32,33,35
A051039 ,1,2,4,8,16,31,46,61,76,91,106,121,136,151,166,181,196,211,226,241,256,271,286,301,316,331,346,361,376,391,406,421,436,451,466,481,496,511,526,541,556,571,586,601,616,631,646,661,676,691,706,721,736,751,
A051040 ,1,2,4,8,16,32,63,94,125,156,187,218,249,280,311,342,373,404,435,466,497,528,559,590,621,652,683,714,745,776,807,838,869,900,931,962,993,1024,1055,1086,1117,1148,1179,1210,1241,1272,1303,1334,1365,1396,1427,
A051041 ,1,4,12,36,96,264,696,1848,4848,12768,33480,87936,230520,604608,1585128,4156392,10895952,28566216,74887056,196322976,514662960,1349208600,3536962584,9272217936,24307198464,63721617888,167046745992,437914664688,1147996820376,3009483583056,7889385389784,20682088837608,54218261608896,
-A051042 ,1,3,9,24,66,180,486,1314,3558,9606,25956,70134,189462,511866,1382880,3735888,10092762,27266340,73661610,
-A051043 ,4,16,60,228,864,3264,12336,46632,176208,665892,2516412,9509364,35935476,135798588,513176076,
+A051042 ,1,3,9,24,66,180,486,1314,3558,9606,25956,70134,189462,511866,1382880,3735888,10092762,27266340,73661610,199001490,537615066,1452399978,3923748270,
+A051043 ,1,4,16,60,228,864,3264,12336,46632,176208,665892,2516412,9509364,35935476,135798588,513176076,1939267560,7328398344,
A051044 ,1,1,3,5,15,27,89,165,585,1113,4097,7917,29927,58499,225585,444793,1741521,3457027,13699699,27342421,109420549,219358315,884987529,1780751883,7233519619,14600965705,59656252987,120742510607,
A051045 ,1,2,8,44,298,2350,22774,
A051046 ,1,109,113,114,199,200,201,241,242,271,277,281,282,283,284,285,286,287,288,289,293,294,295,313,317,318,319,443,444,445,449,450,451,457,458,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,
@@ -53384,7 +53384,7 @@ A053379 ,8,88,888,7888,77888,877888,7877888,87877888,787877888,8787877888,887878
A053380 ,8,88,888,9888,89888,989888,9989888,89989888,989989888,8989989888,98989989888,898989989888,8898989989888,98898989989888,998898989989888,8998898989989888,98998898989989888,898998898989989888,
A053381 ,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,11,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,9,1,3,1,7,1,3,1,8,1,3,1,7,1,3,1,11,1,3,1,7,1,3,
A053382 ,1,1,-1,1,-1,1,1,-3,1,0,1,-2,1,0,-1,1,-5,5,0,-1,0,1,-3,5,0,-1,0,1,1,-7,7,0,-7,0,1,0,1,-4,14,0,-7,0,2,0,-1,1,-9,6,0,-21,0,2,0,-3,0,1,-5,15,0,-7,0,5,0,-3,0,5,1,-11,55,0,-11,0,11,0,-11,0,5,0,1,-6,11,0,-33,0,22,0,
-A053383 ,1,1,2,1,1,6,1,2,2,1,1,1,1,1,30,1,2,3,1,6,1,1,1,2,1,2,1,42,1,2,2,1,6,1,6,1,1,1,3,1,3,1,3,1,30,1,2,1,1,5,1,1,1,10,1,1,1,2,1,1,1,1,1,2,1,66,1,2,6,1,1,1,1,1,2,1,6,1,1,1,1,1,2,1,1,1,2,1,1,1,2730,1,2,1,1,6,1,7,1,10,
+A053383 ,1,1,2,1,1,6,1,2,2,1,1,1,1,1,30,1,2,3,1,6,1,1,1,2,1,2,1,42,1,2,2,1,6,1,6,1,1,1,3,1,3,1,3,1,30,1,2,1,1,5,1,1,1,10,1,1,1,2,1,1,1,1,1,2,1,66,1,2,6,1,1,1,1,1,2,1,6,1,1,1,1,1,2,1,1,1,2,1,1,1,2730,1,2,1,1,6,1,7,1,10,1,3,1,210,1,
A053384 ,2,2,2,2,3,3,3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,5,5,5,5,2,2,2,2,3,3,3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,6,6,6,6,2,2,2,2,3,3,3,3,2,2,2,2,4,4,4,4,2,2,2,2,3,3,3,3,2,2,2,2,5,5,5,5,2,2,2,2,3,3,3,3,2,
A053385 ,0,1,0,3,0,1,0,3,0,1,0,4,0,1,0,4,0,1,0,3,0,1,0,3,0,1,0,5,0,1,0,5,0,1,0,3,0,1,0,3,0,1,0,4,0,1,0,4,0,1,0,3,0,1,0,3,0,1,0,6,0,1,0,6,0,1,0,3,0,1,0,3,0,1,0,4,0,1,0,4,0,1,0,3,0,1,0,3,0,1,0,5,0,1,0,5,0,1,0,3,0,1,0,3,0,
A053386 ,1,1,3,3,1,1,3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,5,5,1,1,5,5,1,1,3,3,1,1,3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,6,6,1,1,6,6,1,1,3,3,1,1,3,3,1,1,4,4,1,1,4,4,1,1,3,3,1,1,3,3,1,1,5,5,1,1,5,5,1,1,3,3,1,1,3,3,1,
@@ -54591,7 +54591,7 @@ A054586 ,1,-1,-3,-5,-1,-9,-11,3,-15,-17,5,-21,-3,-1,-27,-29,9,15,-35,11,-39,-41,
A054587 ,5,37,73,683,631,1459,2633,4621,4733,11093,11527,16193,38993,34183,25189,99929,100069,78941,147881,140071,168151,358109,382117,361111,566567,783421,448249,1050083,1031923,631061,2060687,1843357,1314701,
A054588 ,2,6,14,8,12,54,30,22,14,30,90,20,90,76,90,78,190,60,62,104,186,204,190,96,44,168,254,108,188,80,38,290,174,258,98,44,170,136,132,176,180,156,292,190,312,156,142,158,450,120,130,350,132,610,384,392,430,410,
A054589 ,1,1,1,2,4,3,6,18,25,15,24,96,190,210,105,120,600,1526,2380,2205,945,720,4320,13356,26488,34650,27720,10395,5040,35280,128052,305620,507430,575190,405405,135135,
-A054590 ,0,1,3,19,244,10101,1562298,885237542,1795141933300,13031553571814674,341286507770733602176,32523592049568306757117737,11366810480400463340177768296746,
+A054590 ,0,1,3,19,244,10101,1562298,885237542,1795141933300,13031553571814674,341286507770733602176,32523592049568306757117737,11366810480400463340177768296746,14669108426561606778443288692015619955,70315685953531425166863071956073529852161120,
A054591 ,1,3,4,9,10,12,13,27,28,30,36,39,40,81,82,84,90,91,108,117,120,121,146,182,205,243,244,246,252,270,273,324,328,351,360,363,364,386,438,546,615,656,671,729,730,732,738,756,757,810,819,820,949,972,984,
A054592 ,0,1,4,26,296,6064,230896,16886864,2423185664,687883494016,387139470010624,432380088071584256,959252253993204724736,4231267540316814507357184,37138269572860613284747227136,
A054593 ,0,1,10,262,21496,6433336,7566317200,35247649746352,648839620390462336,47230175230392839683456,13617860445102311268975051520,15577054031612736747163633737901312,
@@ -54641,7 +54641,7 @@ A054636 ,0,1,3,6,10,15,21,28,29,29,30,31,32,34,35,38,39,43,44,49,50,56,57,64,66,
A054637 ,0,1,3,4,4,5,6,7,9,11,11,13,14,16,18,19,19,19,20,20,21,22,22,24,25,26,26,27,28,29,30,31,33,34,36,36,37,39,40,41,43,45,47,47,47,49,49,50,52,52,54,56,57,57,59,60,61,63,64,66,68,70,70,72,74,75,
A054638 ,0,1,1,1,0,0,1,0,0,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,1,1,
A054639 ,1,2,3,5,6,9,11,14,18,23,26,29,30,33,35,39,41,50,51,53,65,69,74,81,83,86,89,90,95,98,99,105,113,119,131,134,135,146,155,158,173,174,179,183,186,189,191,194,209,210,221,230,231,233,239,
-A054640 ,3,12,72,576,6912,96768,1741824,34836480,836075520,25082265600,802632499200,30500034969600,1281001468723200,56364064623820800,2705475101943398400,146095655504943513600,8765739330296610816000,543475838478389870592000,36956357016530511200256000,
+A054640 ,1,3,12,72,576,6912,96768,1741824,34836480,836075520,25082265600,802632499200,30500034969600,1281001468723200,56364064623820800,2705475101943398400,146095655504943513600,8765739330296610816000,543475838478389870592000,36956357016530511200256000,
A054641 ,1,6,6,6,6,42,42,210,210,210,210,3990,3990,43890,43890,43890,43890,1360590,23130030,23130030,855811110,855811110,855811110,855811110,855811110,855811110,11125544430,11125544430,11125544430,11125544430,
A054642 ,1,6,42,210,3990,43890,1360590,23130030,855811110,11125544430,255887521890,20215114229310,828819683401710,24035770818649590,2331469769409010230,123567897778677542190,5313419604483134314170,
A054643 ,3,47,151,167,199,251,257,367,503,523,557,587,601,647,727,941,971,991,1063,1097,1117,1181,1217,1231,1361,1453,1493,1499,1531,1741,1747,1753,1759,1889,1901,1907,2063,2161,2281,2393,2399,2411,2441,2671,2897,2957,
@@ -54740,7 +54740,7 @@ A054735 ,8,12,24,36,60,84,120,144,204,216,276,300,360,384,396,456,480,540,564,62
A054736 ,4,8,11,15,21,29,40,55,76,
A054737 ,253,1265,2225,2530,6325,12650,18025,18975,22250,24982,25300,
A054738 ,64009,1600225,4950625,6400900,40005625,160022500,324900625,360050625,495062500,624100324,640090000,
-A054739 ,1,3,21,2862,5398083,105918450471,18761832172500795,29912416165371498901002,429210477536602279123636967061,55428311030379722725246681652572022523,
+A054739 ,1,3,21,2862,5398083,105918450471,18761832172500795,29912416165371498901002,429210477536602279123636967061,55428311030379722725246681652572022523,64422190091501416379601522735200323789074174081,673878862467911703904942451533575765568815772023224550102,
A054740 ,0,1,1,2,1,2,1,2,2,1,2,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,
A054741 ,6,10,12,14,18,20,22,24,26,28,30,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,105,106,108,110,112,114,116,118,120,122,124,126,130,132,134,136,
A054742 ,2,41,1952,172043,20511924,3058135804,545880769246,113492835877474,26936031159146324,7186257876123323136,2129016419091882758064,693526953186674417975860,246375213208005330322801608,94795009032593187381371471299,39271207630529921493096501099998,17428450442901657489782698628853383,8249301503003544171210026750727519638,
@@ -54752,8 +54752,8 @@ A054747 ,3,76,4003,352744,41876694,6217447912,1106509486839,229553329028386,5439
A054748 ,1,6,29,209,1652,15981,171837,2060481,26951143,381398614,
A054749 ,1,5,23,165,1328,13193,145076,1774515,23600723,338704176,
A054750 ,2,3,5,7,29,67,89,199,599,2999,4999,29989,59999,79999,389999,989999,6999899,8989999,59899999,89999999,289999999,799999999,3999998999,19999997999,79999999999,399999998999,599999899999,999998999999,
-A054751 ,1,4,55,34960,537157696,140738033618944,590295811483987148800,39614081257168338331296071680,42535295865117309120430975675097153536,730750818665451459102461990840694008379514814464,
-A054752 ,1,5,120,252375,19076074375,37252918396015625,1818989403666496277343750,2220446049250331744551658935546875,67762635780344027129112510010600128173828125,
+A054751 ,1,4,55,34960,537157696,140738033618944,590295811483987148800,39614081257168338331296071680,42535295865117309120430975675097153536,730750818665451459102461990840694008379514814464,200867255532373784442745261867639247948787687313041365401600,
+A054752 ,1,5,120,252375,19076074375,37252918396015625,1818989403666496277343750,2220446049250331744551658935546875,67762635780344027129112510010600128173828125,51698788284564229679463057470911735435947895050048828125,
A054753 ,12,18,20,28,44,45,50,52,63,68,75,76,92,98,99,116,117,124,147,148,153,164,171,172,175,188,207,212,236,242,244,245,261,268,275,279,284,292,316,325,332,333,338,356,363,369,387,388,404,412,423,425,428,436,452,
A054754 ,1,2,5,8,17,32,37,101,125,128,197,257,401,468,512,577,677,1297,1417,1601,1872,2048,2340,2917,3125,3137,3145,4100,4212,4357,4913,5477,7057,7488,8101,8192,8837,9360,12101,13457,14401,14841,15377,15588,15877,
A054755 ,2,5,8,17,32,37,101,125,128,197,257,401,512,577,677,1297,1601,2048,2917,3125,3137,4357,4913,5477,7057,8101,8192,8837,12101,13457,14401,15377,15877,16901,17957,21317,22501,24337,25601,28901,30977,32401,
@@ -56279,9 +56279,9 @@ A056274 ,1,1,4,12,40,116,364,1080,3276,9800,29524,88440,265720,796796,2391440,71
A056275 ,1,1,4,13,50,181,714,2780,11046,43895,175274,699875,2798250,11188191,44747380,178970560,715860650,2863365834,11453377194,45813202675,183252461532,733008625151,2932033104554,11728127521060,
A056276 ,1,1,4,13,51,196,854,3830,17997,86419,422004,2079260,10306751,51263086,255514299,1275160060,6368612301,31821454413,159042661904,795019250650,3974515029793,19870830290476,99348921288654,
A056277 ,1,1,4,13,51,197,875,4096,20643,109246,601491,3402911,19628063,114699438,676207572,4010086352,23874362199,142508702805,852124263683,5101098123207,30560194492576,183176169456214,
-A056278 ,0,1,3,6,15,27,63,120,252,495,1023,2010,4095,8127,16365,32640,65535,130788,262143,523770,1048509,2096127,4194303,8386440,16777200,33550335,67108608,134209530,268435455,
-A056279 ,0,0,1,6,25,89,301,960,3024,9305,28501,86430,261625,788669,2375075,7140720,21457825,64435896,193448101,580597110,1742343323,5228050949,15686335501,47063113320,141197991000,
-A056280 ,0,0,0,1,10,65,350,1700,7770,34095,145750,611435,2532530,10391395,42355940,171797200,694337290,2798799150,11259666950,45232081795,181509069700,727778478075,2916342574750,11681056021300,
+A056278 ,0,1,3,6,15,27,63,120,252,495,1023,2010,4095,8127,16365,32640,65535,130788,262143,523770,1048509,2096127,4194303,8386440,16777200,33550335,67108608,134209530,268435455,536854005,1073741823,2147450880,4294966269,8589869055,17179869105,
+A056279 ,0,0,1,6,25,89,301,960,3024,9305,28501,86430,261625,788669,2375075,7140720,21457825,64435896,193448101,580597110,1742343323,5228050949,15686335501,47063113320,141197991000,423610488665,1270865802276,3812663735790,11438127792025,34314649427035,
+A056280 ,0,0,0,1,10,65,350,1700,7770,34095,145750,611435,2532530,10391395,42355940,171797200,694337290,2798799150,11259666950,45232081795,181509069700,727778478075,2916342574750,11681056021300,46771289738800,187226354413735,749329038527580,
A056281 ,0,0,0,0,1,15,140,1050,6951,42524,246730,1379385,7508501,40074895,210766919,1096189500,5652751651,28958088579,147589284710,749206047975,3791262568261,19137821665325,96416888184100,
A056282 ,0,0,0,0,0,1,21,266,2646,22827,179487,1323651,9321312,63436352,420693273,2734926292,17505749898,110687248392,693081601779,4306078872557,26585679462783,163305339165738,998969857983405,
A056283 ,0,0,2,9,30,91,258,729,2018,5613,15546,43315,120750,338259,950062,2678499,7573350,21480739,61088874,174184755,497812638,1425847623,4092087522,11765822365,33887517870,97756387365,282414624746,816999710223,2366509198350,6862930841141,
@@ -56851,7 +56851,7 @@ A056846 ,1,2,11,80,780,8781,104828,1298506,16462696,212457221,2780615627,3681703
A056847 ,0,0,1,1,2,3,4,4,5,6,7,8,9,9,10,11,12,13,14,15,16,16,17,18,19,20,21,22,23,24,25,25,26,27,28,29,30,31,32,33,34,35,36,36,37,38,39,40,41,42,43,44,45,46,47,48,49,49,50,51,52,53,54,55,56,57,58,59,
A056848 ,1,10,16,65,160,180,366,406,896,1436,3904,5464,6312,7168,12558,17957,36960,48097,48256,61952,88646,94400,107340,112240,114863,127540,171856,270336,383360,392736,459012,623639,960484,1222656,1312768,1463990,1480704,2244736,2380968,3183563,4161888,4787280,5107455,5606400,6826556,7878400,9188414,9533238,10219520,10356472,12981760,15162808,22062080,25240360,28313472,32215040,41284864,72160576,79563520,91164167,
A056849 ,1,4,7,6,5,6,3,6,9,0,1,6,3,6,5,6,7,4,9,0,1,4,7,6,5,6,3,6,9,0,1,6,3,6,5,6,7,4,9,0,1,4,7,6,5,6,3,6,9,0,1,6,3,6,5,6,7,4,9,0,1,4,7,6,5,6,3,6,9,0,1,6,3,6,5,6,7,4,9,0,1,4,7,6,5,6,3,6,9,0,1,6,3,6,5,6,7,4,9,0,
-A056850 ,0,1,1,5,17,13,217,139,1631,3299,6487,46075,7153,502829,588665,2428309,9492289,5077565,118985033,88519643,808182895,1870418611,2978678759,25423702091,7551629537,252223018333,342842572777,1170495537221,
+A056850 ,0,1,1,5,17,13,217,139,1631,3299,6487,46075,7153,502829,588665,2428309,9492289,5077565,118985033,88519643,808182895,1870418611,2978678759,25423702091,7551629537,252223018333,342842572777,1170495537221,5284606410545,1738366812781,
A056851 ,0,1,2,3,11,26,83,128,186,258,572,875,1494,2029,3859,4810,6497,9274,18033,19243,24600,26073,30828,32528,34287,41930,48325,96475,103590,118814,126936,205022,240742,260009,331334,379612,396656,405360,414186,
A056852 ,7,521,102943,23775972551,21633936185161,45957792327018709121,98920982783015679456199,870019499993663001431459704607,85589538438707037818727607157700537549449,
A056853 ,4,6,7,12,13,15,18,19,21,30,42,45,60,63,72,93,102,108,117,138,150,165,180,192,198,213,228,240,255,270,282,312,333,348,357,420,432,453,462,522,525,570,600,618,642,660,693,717,765,810,822,828,858,882,933,957,
@@ -57753,7 +57753,7 @@ A057748 ,0,1,2,2,2,3,4,3,3,5,4,3,4,3,2,2,3,5,5,2,2,5,4,2,4,7,7,6,5,4,4,3,3,5,6,6
A057749 ,13,19,37,43,53,59,61,67,83,101,107,109,131,139,149,157,163,173,179,181,197,211,227,229,251,269,277,283,293,307,311,317,331,347,349,373,379,389,397,419,421,443,461,467,491,499,509,523,541,547,557,563,571,
A057750 ,0,1,4,10,23,49,100,202,413,839,1713,3493,7130,14535,29617,60158,122077,247132,499409,1007440,2029801,4083888,8208828,16484742,
A057751 ,2,3,5,7,11,17,23,29,31,41,47,71,73,79,89,97,103,113,127,137,151,167,191,193,199,223,233,239,241,257,263,271,281,313,337,353,359,367,383,401,409,431,433,439,449,457,463,479,487,503,521,569,577,593,599,601,
-A057752 ,2,5,10,17,38,130,339,754,1701,3104,11588,38263,108971,314890,1052619,3214632,7956589,21949555,99877775,222744644,597394254,1932355208,7250186216,17146907278,55160980939,155891678121,508666658006,
+A057752 ,2,5,10,17,38,130,339,754,1701,3104,11588,38263,108971,314890,1052619,3214632,7956589,21949555,99877775,222744644,597394254,1932355208,7250186216,17146907278,55160980939,155891678121,508666658006,1427745660374,
A057753 ,1,2,10,27,150,641,3796,
A057754 ,6,30,178,1246,9630,78628,664918,5762209,50849235,455055615,4118066401,37607950281,346065645810,3204942065692,29844571475288,279238344248557,2623557165610822,24739954309690415,234057667376222382,
A057755 ,1,1,2,3,5,10,20,39,78,155,309,617,1234,2467,4933,9865,19729,39457,78914,157827,315653,631306,1262612,2525223,5050446,10100891,20201782,40403563,80807125,161614249,323228497,646456994,1292913987,2585827973,
@@ -57795,7 +57795,7 @@ A057790 ,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,2,3,1,1,1,1,3,4,4,3,1,1
A057791 ,1,1,3,4,22,6,209,376,1835,2540,49863,94944,1151914,2190666,12079274,95722288,1150653920,3217888350,47454745803,130819911320,846278385786,8064305838350,126356632390297,288019285668096,6861189820377586,
A057792 ,1,1,5,28,288,3126,50069,826696,17604145,388244060,10405071317,285312497280,9211817190184,303160805686506,11415167261421900,438197051187369424,18896062057839751444,827240565046755853710,
A057793 ,5,26,168,1227,9587,78527,664667,5761552,50847455,455050683,4118052495,37607910542,346065531066,3204941731602,29844570495887,279238341360977,2623557157055978,24739954284239494,234057667300228940,
-A057794 ,1,1,0,-2,-5,29,88,97,-79,-1828,-2318,-1476,-5773,-19200,73218,327052,-598255,-3501366,23884333,-4891825,-86432204,-127132665,1033299853,-1658989719,
+A057794 ,1,1,0,-2,-5,29,88,97,-79,-1828,-2318,-1476,-5773,-19200,73218,327052,-598255,-3501366,23884333,-4891825,-86432204,-127132665,1033299853,-1658989719,-1834784714,-17149335456,-17535487934,-174760519827,
A057795 ,1,1,3,7,33,121,873,5167,45507,367927,4037913,39921961,522956313,6267300607,93445274187,1313941673647,22324392524313,355693695038761,6780385526348313,122000794104233527,2554923725074062867,
A057796 ,672,2552,31552,12672,14472,62232,355080,95040,1225008,
A057797 ,672,2552,43696,14472,528,62232,407880,42240,1225008,
@@ -57840,7 +57840,7 @@ A057835 ,0,3,23,143,906,6116,44158,332774,2592592,20758029,169923159,1416705193,
A057836 ,22,38,58,62,82,85,87,95,98,115,118,122,123,138,142,158,162,175,178,182,185,202,203,207,213,215,217,218,220,238,242,258,262,265,275,278,284,297,298,302,318,322,325,333,335,338,355,357,358,362,365,370,371,
A057837 ,1,0,0,0,1,1,1,1,36,127,337,793,7525,48764,238954,997790,6401435,49107697,345482807,2150694855,14656830110,116678887407,978172378669,7886661080873,63905475745765,553437891603452,5122279358273976,
A057838 ,2,3,11,35,71,191,419,659,1091,1199,1379,1655,2015,2135,2339,2591,3059,4439,6119,6215,6335,7055,8099,8351,8519,9815,11159,12419,12431,12599,12719,12851,13679,15119,15239,16415,16919,17255,17879,18215,18479,
-A057839 ,1,31,198089,
+A057839 ,1,31,198089,876881261,2026728077,
A057840 ,1,3,7,9,27,49,81,133,243,267,343,729,2187,2401,5999,6561,14063,14337,16807,17253,19683,22329,33323,45619,59049,75573,117649,144531,177147,348519,383913,531441,745339,823543,911853,
A057841 ,4,8,2048,34359738368,2361183241434822606848,3138550867693340381917894711603833208051177722232017256448,
A057842 ,1,13,4381,560129,606797,
@@ -58255,7 +58255,7 @@ A058250 ,1,1,2,2,6,30,30,30,30,330,2310,2310,2310,2310,2310,53130,690690,2003001
A058251 ,2,6,120,1680,36960,5765760,1568286720,536354058240,24672286679040,2861985254768640,2661646286934835200,3545312854197200486400,5814313080883408797696000,10500649424075436288638976000,
A058252 ,5321191,8606621,9148351,41675791,43251251,49820291,51825461,57791281,66637721,73114441,74055851,82584841,86801801,87620011,112161451,123720361,125810021,126265751,136413721,140969291,152777291,153348161,
A058253 ,2,3,5,7,11,47,61,347,7057,
-A058254 ,1,2,4,12,60,60,240,720,7920,55440,55440,55440,55440,55440,1275120,16576560,480720240,480720240,480720240,480720240,480720240,480720240,19709529840,19709529840,39419059680,197095298400,3350620072800,
+A058254 ,1,1,2,4,12,60,60,240,720,7920,55440,55440,55440,55440,55440,1275120,16576560,480720240,480720240,480720240,480720240,480720240,480720240,19709529840,19709529840,39419059680,197095298400,3350620072800,177582863858400,532748591575200,
A058255 ,1,2,4,12,60,240,720,7920,55440,1275120,16576560,480720240,19709529840,39419059680,197095298400,3350620072800,177582863858400,532748591575200,19711697888282400,59135093664847200,
A058256 ,2,2,3,5,1,4,3,11,7,1,1,1,1,23,13,29,1,1,1,1,1,41,1,2,5,17,53,3,1,1,1,1,1,37,1,1,3,83,43,89,1,19,2,7,1,1,1,113,1,1,1,1,5,4,131,67,1,1,1,47,73,1,31,1,79,1,1,173,1,1,179,61,1,1,191,97,1,1,1,1,1,1,1,1,1,1,1,1,1,
A058257 ,1,0,1,0,0,1,1,1,1,0,3,2,1,0,0,0,3,5,6,6,6,0,0,3,8,14,20,26,71,71,71,68,60,46,26,0,413,342,271,200,132,72,26,0,0,0,413,755,1026,1226,1358,1430,1456,1456,1456,0,0,413,1168,2194,3420,4778,6208,7664,9120,10576,
@@ -60868,7 +60868,7 @@ A060863 ,2,3,4,5,6,7,9,11,12,13,15,17,18,19,21,22,23,26,29,30,31,34,37,38,39,41,
A060864 ,1,8,10,14,16,20,24,25,27,28,32,33,35,36,40,44,48,54,57,58,62,63,65,66,74,75,80,84,85,88,90,94,98,104,118,119,121,128,136,140,141,142,146,147,148,152,156,158,159,161,162,164,168,171,172,174,178,182,184,188,
A060865 ,1,1,2,1,1,8,1,1,2,1,1,16,1,1,2,1,1,8,1,1,2,1,1,32,1,1,2,1,1,8,1,1,2,1,1,16,1,1,2,1,1,8,1,1,2,1,1,64,1,1,2,1,1,8,1,1,2,1,1,16,1,1,2,1,1,8,1,1,2,1,1,32,1,1,2,1,1,8,1,1,2,1,1,16,1,1,2,1,1,8,1,1,2,1,1,128,1,1,2,1,
A060866 ,2,3,4,9,6,12,8,15,16,18,12,28,14,24,24,35,18,39,20,42,32,36,24,60,36,42,40,56,30,72,32,63,48,54,48,97,38,60,56,90,42,96,44,84,78,72,48,124,64,93,72,98,54,120,72,120,80,90,60,168,62,96,104,135,84,144,68,126,96,
-A060867 ,1,9,49,225,961,3969,16129,65025,261121,1046529,4190209,16769025,67092481,268402689,1073676289,4294836225,17179607041,68718952449,274876858369,1099509530625,4398042316801,17592177655809,
+A060867 ,1,9,49,225,961,3969,16129,65025,261121,1046529,4190209,16769025,67092481,268402689,1073676289,4294836225,17179607041,68718952449,274876858369,1099509530625,4398042316801,17592177655809,70368727400449,281474943156225,1125899839733761,
A060868 ,2,32,338,3200,29282,264992,2389298,21516800,193690562,1743333152,15690352658,141214236800,1270931319842,11438391444512,102945551698418,926510051379200,8338590720693122,75047317261079072,675425857674234578,
A060869 ,3,75,1323,21675,348843,5589675,89467563,1431612075,22906317483,366503176875,5864059218603,93824981052075,1501199831050923,24019197833685675,384307167486454443,6148914688373205675,98382635048331029163,1574122160910735420075,
A060870 ,4,144,3844,97344,2439844,61027344,1525839844,38146777344,953673339844,23841853027344,596046423339844,14901161071777344,372529029235839844,9313225743103027344,232830643638610839844,5820766091270446777344,
@@ -60921,7 +60921,7 @@ A060916 ,101,127,131,149,151,163,167,181,191,307,311,421,431,433,457,461,479,487
A060917 ,1,12,150,2180,36855,715008,15697948,385300800,10463945085,311697869120,10108450408914,354630018043392,13384651003544275,540860323696035840,23300648262667635960,1066165291831917811712,
A060918 ,1,20,360,6860,143570,3321864,84756000,2372001720,72384192540,2394775746220,85443353291296,3271908306712500,133893717061821080,5832748749666611920,269542701201588099840,13172225935626444660144,678788199609330554538000,36790272488566573278647940,
A060919 ,4,8,20,60,204,748,2860,11180,44204,175788,701100,2800300,11193004,44755628,178989740,715893420,2863442604,11453508268,45813508780,183252986540,733009849004,2932035201708,11728132418220,46912512895660,
-A060920 ,1,2,1,5,5,1,13,20,9,1,34,71,51,14,1,89,235,233,105,20,1,233,744,942,594,190,27,1,610,2285,3522,2860,1295,315,35,1,1597,6865,12473,12402,7285,2534,490,44,1,4181,20284,
+A060920 ,1,2,1,5,5,1,13,20,9,1,34,71,51,14,1,89,235,233,105,20,1,233,744,942,594,190,27,1,610,2285,3522,2860,1295,315,35,1,1597,6865,12473,12402,7285,2534,490,44,1,4181,20284,42447,49963,36122,16407,4578,726,54,1,
A060921 ,1,3,2,8,10,3,21,38,22,4,55,130,111,40,5,144,420,474,256,65,6,377,1308,1836,1324,511,98,7,987,3970,6666,6020,3130,924,140,8,2584,11822,23109,25088,16435,6588,1554,192,9,
A060922 ,1,3,1,4,6,1,7,17,9,1,11,38,39,12,1,18,80,120,70,15,1,29,158,315,280,110,18,1,47,303,753,905,545,159,21,1,76,566,1687,2568,2120,942,217,24,1,123,1039,3612,6666,7043,4311,
A060923 ,1,4,1,11,17,1,29,80,39,1,76,303,315,70,1,199,1039,1687,905,110,1,521,3364,7470,6666,2120,159,1,1364,10493,29634,37580,20965,4311,217,1,3571,31885,109421,181074,148545,
@@ -60945,7 +60945,7 @@ A060940 ,2,3,7,5,5,7,11,7,13,19,7,11,29,23,17,11,19,13,17,11,13,23,19,11,13,23,4
A060941 ,1,2,23,377,7229,151491,3361598,77635093,1846620581,44930294909,1113015378438,27976770344941,711771461238122,18293652115906958,474274581883631615,12388371266483017545,325714829431573496525,8613086428709348334675,228925936056388155632081,
A060942 ,2,12,420,27720,
A060943 ,1,5,251,357904,25795462624,141727869124448256,83296040059942781485105152,7013444132843374500928464765799366656,109329825340451764123791003609208862665771818418176,396334659032531033249146049131230887376087800711479296000000000000,
-A060944 ,1,9,130,2900,93576,4141872,241353792,17929776384,1655071418880,185914776960000,24978180045312000,3955930130221056000,729464836964806656000,154952762244805582848000,
+A060944 ,1,9,130,2900,93576,4141872,241353792,17929776384,1655071418880,185914776960000,24978180045312000,3955930130221056000,729464836964806656000,154952762244805582848000,37566943754471090749440000,10310706109241121091092480000,
A060945 ,1,1,2,3,6,10,18,31,55,96,169,296,520,912,1601,2809,4930,8651,15182,26642,46754,82047,143983,252672,443409,778128,1365520,2396320,4205249,7379697,12950466,22726483,39882198,69988378,122821042,215535903,378239143,663763424,1164823609,
A060946 ,1,3,12,76,701,8477,126126,2223278,45269999,1045269999,26982694600,769991065288,24068076187769,817782849441913,30010708874832538,1182932213481679514,49844124089148547995,2235755683827845079963,106363105981739086612804,
A060947 ,513,561,585,633,645,693,717,765,771,819,843,891,903,951,975,1023,19684,20008,20332,20440,20764,21088,21196,21520,21844,21880,22204,22528,22636,22960,23284,23392,23716,24040,24076,24400,24724,24832,
@@ -60985,7 +60985,7 @@ A060980 ,309,408,419,507,518,529,606,617,628,639,705,716,727,738,749,804,815,826
A060981 ,1373,1447,1097,1163,853,911,641,691,461,503,313,347,197,223,113,131,61,71,41,43,53,47,97,83,173,151,281,251,421,383,593,547,797,743,1033,971,1301,1231,1601,1523,1933,1847,2297,2203,2693,2591,3121,3011,3581,
A060982 ,11,10,13,14,15,16,17,18,19,90,109,209,309,409,509,609,709,809,909,10909,20909,30909,40909,50909,60909,70909,80909,90909,1090909,2090909,3090909,4090909,5090909,6090909,7090909,8090909,9090909,109090909,
A060983 ,1,7,13,35,31,91,57,154,130,217,133,455,183,399,403,644,307,910,381,1085,741,931,553,2002,806,1281,1209,1995,871,2821,993,2632,1729,2149,1767,4550,1407,2667,2379,4774,1723,5187,1893,4655,4030,3871,
-A060984 ,1,2,3,4,8,12,21,37,73,137,258,514,998,1959,3895,7739,15308,30437,60713,121229,242333,484397,967422,1933711,3865811,7730967,15459367,30912128,61814609,123625653,247235577,494448306,988888002,1977738918,3955408759,
+A060984 ,1,2,3,4,8,12,21,37,73,137,258,514,998,1959,3895,7739,15308,30437,60713,121229,242333,484397,967422,1933711,3865811,7730967,15459367,30912128,61814609,123625653,247235577,494448306,988888002,1977738918,3955408759,7910812423,
A060985 ,1,2,3,6,12,22,43,79,157,310,610,1205,2381,4727,9383,18699,37227,74355,148660,296900,593735,1187240,2373810,4746741,9491481,18981027,37956907,75910735,151820416,303627016,607253419,1214497244,2428978214,4857918665,
A060986 ,2,3,5,7,12,19,31,34,53,87,118,205,323,441,259,612,730,
A060987 ,1,2,3,4,7,11,18,20,31,51,69,120,189,258,327,358,427,
@@ -61177,7 +61177,7 @@ A061172 ,9,120,753,3612,15040,57366,206115,709152,2360943,7659870,24340184,76031
A061173 ,3,70,642,4055,20945,95372,398290,1561683,5836190,21001410,73300478,249476600,831342517,2720979140,8768966810,27881856697,87610794135,272424413508,839229287580,2563768831145,7773145679478,
A061174 ,15,545,7043,57560,365045,1970905,9520315,42385132,177293730,705980760,2701362950,10001654350,36020160943,126701700755,436709397085,1478813477920,4930328078835,16212542696607,
A061175 ,9,471,8268,85962,662773,4215123,23440212,118073914,551281476,2423731704,10148667670,40812739230,158644493079,599051383561,2206150654944,7949311477362,28098758599203,97645872621753,
-A061176 ,1,1,-1,1,-1,1,1,0,0,-1,1,2,-5,2,1,1,5,-15,15,-5,-1,1,9,-30,41,-30,9,1,1,14,-49,77,-77,49,-14,-1,1,20,-70,112,-125,112,-70,20,1,1,27,-90,126,-117,117,-126,90,-27,-1,1,35,-105,90,45,-131,45,
+A061176 ,1,1,-1,1,-1,1,1,0,0,-1,1,2,-5,2,1,1,5,-15,15,-5,-1,1,9,-30,41,-30,9,1,1,14,-49,77,-77,49,-14,-1,1,20,-70,112,-125,112,-70,20,1,1,27,-90,126,-117,117,-126,90,-27,-1,1,35,-105,90,45,-131,45,90,-105,35,1,
A061177 ,1,2,-2,3,-5,3,4,-8,8,-4,5,-10,11,-10,5,6,-10,6,-6,10,-6,7,-7,-14,29,-14,-7,7,8,0,-56,120,-120,56,0,-8,9,12,-126,288,-365,288,-126,12,9,10,30,-228,540,-770,770,-540,228,-30,-10,11,55,-363,858,
A061178 ,1,9,51,233,942,3522,12473,42447,140109,451441,1426380,4434420,13599505,41225349,123723351,368080793,1086665562,3186317718,9286256393,26916587307,77634928209,222920650081,
A061179 ,1,14,105,594,2860,12402,49963,190570,696787,2463300,8472280,28481220,93914325,304597382,973877245,3075011478,9602753412,29695165110,91026167999,276833858530,835933445799,2507876305416,
@@ -61540,7 +61540,7 @@ A061535 ,2,3,5,8,12,17,24,33,44,58,74,95,119,149,184,226,274,332,399,477,568,671
A061536 ,1,2,4,6,9,12,16,20,24,28,33,38,44,50,56,62,69,76,84,92,100,108,117,126,135,144,153,162,172,182,193,204,215,226,237,248,260,272,284,296,309,322,336,350,364,378,393,408,423,438,453,468,484,500,516,532,548,
A061537 ,1,2,3,4,5,36,7,8,9,100,11,144,13,196,225,16,17,324,19,400,441,484,23,576,25,676,27,784,29,810000,31,32,1089,1156,1225,1296,37,1444,1521,1600,41,3111696,43,1936,2025,2116,47,2304,49,2500,2601,2704,53,2916,
A061538 ,1,1,1,2,1,1,1,8,3,1,1,12,1,1,1,64,1,18,1,20,1,1,1,576,5,1,27,28,1,1,1,1024,1,1,1,7776,1,1,1,1600,1,1,1,44,45,1,1,110592,7,50,1,52,1,2916,1,3136,1,1,1,3600,1,1,63,32768,1,1,1,68,1,1,1,26873856,1,1,75,76,1,1,1,
-A061539 ,2,7,28,116,490,2094,9014,38988,169184,
+A061539 ,1,2,7,28,116,490,2094,9014,38988,169184,735846,3205830,13984076,61057108,266780436,1166320956,5101254296,22319861332,97685806958,427635145446,1872400460940,8199602319764,35912342632908,157304824211156,689096352589448,3018916616772272,
A061540 ,0,0,0,6,205,5700,156555,4483360,136368414,4432075200,154060613850,5720327205120,226378594906035,9523895202838016,424814409531910125,20037831121798963200,996964614369038858060,52198565072252054814720,
A061541 ,0,0,0,1,120,6165,258125,10230360,405918324,16530124800,699126562530,30884683104000,1428626760992860,69248819808744576,3516693960681822375,186964957159176734720,10395215954531344335000,603712553730550509035520,36575888366817680447745924,
A061542 ,0,0,0,0,45,4945,331506,18602136,974679363,50088981600,2588876118675,136440380444544,7389687834858186,413138671455654144,23901631262740105875,1432747304604594800640,89030607737889046580442,
@@ -61893,7 +61893,7 @@ A061888 ,10,5063,14573,17098,1916357,468726713734,
A061889 ,1,1,2,2,3,2,4,3,5,5,7,8,10,8,13,14,15,18,20,23,29,31,36,41,49,54,63,72,80,92,108,116,137,153,174,197,222,250,281,318,354,398,450,497,561,624,697,779,869,964,1075,1193,1325,1471,1635,1809,2004,2217,2455,2711,
A061890 ,100,25633969,212372329,292341604,3672424151449,219704732167875184222756,
A061891 ,1,1,4,7,7,10,13,13,16,19,19,22,25,25,28,31,31,34,37,37,40,43,43,46,49,49,52,55,55,58,61,61,64,67,67,70,73,73,76,79,79,82,85,85,88,91,91,94,97,97,100,103,103,106,109,109,112,115,115,118,121,121,124,
-A061892 ,0,3,1,3,3,6,10,28,108,1011,32511,9314238,
+A061892 ,0,3,1,3,3,6,10,28,108,1011,32511,9314238,84560776390,
A061893 ,1,0,0,3,1,1,1,1,2,3,1,1,1,2,2,3,1,1,1,2,2,3,3,4,5,6,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,7,7,8,9,10,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,
A061894 ,0,2,2,4,6,13,35,171,1934,97151,52942129,
A061895 ,2,0,2,2,1,1,2,2,1,1,2,2,3,4,1,1,1,1,2,2,2,3,3,4,4,5,6,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,8,8,9,9,10,11,12,13,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,
@@ -62275,7 +62275,7 @@ A062270 ,3,45,175,693,11011,2807805,302307005,402243205,714186915,42803602439,11
A062271 ,4,64,256,1024,16384,4194304,452984832,603979776,1073741824,64424509440,16698832846848,8906044184985600,2244323134616371200,4588393964104581120,24471434475224432640,32628579300299243520,
A062272 ,1,1,2,5,12,41,152,685,3472,19921,126752,887765,6781632,56126201,500231552,4776869245,48656756992,526589630881,6034272215552,72989204937125,929327412759552,12424192360405961,174008703107274752,
A062273 ,1,23,456,7890,12345,678901,2345678,90123456,789012345,6789012345,67890123456,789012345678,9012345678901,23456789012345,678901234567890,1234567890123456,78901234567890123,456789012345678901,
-A062274 ,0,0,1,7,45,291,2030,15695,135045,1287243,13495669,154516663,1919455487,25721712601,
+A062274 ,0,0,1,7,45,291,2030,15695,135045,1287243,13495669,154516663,1919455487,25721712601,369942275033,
A062275 ,1,0,0,0,1,0,0,2,2,0,0,3,16,3,0,0,4,72,72,4,0,0,5,256,729,256,5,0,0,6,800,5184,5184,800,6,0,0,7,2304,30375,65536,30375,2304,7,0,0,8,6272,157464,640000,640000,157464,6272,8,0,0,9,16384,750141,5308416,9765625,
A062276 ,0,0,0,1,1,2,2,2,3,3,3,4,4,4,5,5,6,6,6,7,7,7,8,8,9,9,9,10,10,10,11,11,11,12,12,13,13,13,14,14,14,15,15,16,16,16,17,17,17,18,18,18,19,19,20,20,20,21,21,21,22,22,22,23,23,24,24,24,25,25,25,26,26,27,27,27,28,28,
A062277 ,2,1,1,1,1,3,5,10,20,42,88,189,414,921,2077,4737,10921,25416,59646,141033,335752,804258,1937372,4690989,11412140,27884328,68407056,168446547,416226830,1031816793,2565591729,6397371713,15994440540,
@@ -62448,7 +62448,7 @@ A062443 ,0,0,0,2,8,21,45,86,150,247,387,580,840,1183,1626,2188,2889,3753,4805,60
A062444 ,0,0,1,3,9,21,46,86,151,248,387,580,841,1184,1627,2188,2889,3753,4805,6073,7587,9378,11481,13934,16775,20048,23796,28068,32913,38385,44540,51436,59136,67704,77208,87720,99314,112067,126061,141379,158109,
A062445 ,0,0,0,2,9,23,50,94,164,269,418,623,899,1262,1728,2317,3051,3953,5049,6368,7939,9796,11973,14508,17441,20815,24676,29071,34052,39672,45988,53060,60950,69723,79450,90200,102050,115078,129366,144998,162063,
A062446 ,0,0,1,3,10,24,50,95,165,269,418,624,900,1262,1728,2317,3051,3953,5050,6369,7940,9796,11973,14508,17441,20816,24676,29072,34052,39673,45989,53060,60950,69724,79450,90201,102051,115079,129367,144999,
-A062447 ,3,13,113,1291,20443,350741,7535917,173723909,4735396021,160158887507,5681909839321,237981875869277,10922592672383377,520256404164231967,26879210482658444513,1556951523272630146229,
+A062447 ,2,3,13,113,1291,20443,350741,7535917,173723909,4735396021,160158887507,5681909839321,237981875869277,10922592672383377,520256404164231967,26879210482658444513,1556951523272630146229,
A062448 ,2,2,7,103,1619,28687,567871,12579617,310248241,8448283757,252097800623,8187297276401,287629183566841,10871811891162227,440023773804299591,
A062449 ,0,0,1,7,24,59,123,228,393,634,976,1444,2066,2877,3912,5212,6820,8786,11161,14002,17370,21331,25953,31312,37486,44560,52623,61767,72093,83705,96711,111226,127371,145270,165056,186864,210838,237125,265881,
A062450 ,0,0,2,8,25,59,123,229,393,635,976,1444,2067,2877,3913,5212,6821,8786,11162,14003,17371,21331,25953,31312,37486,44560,52623,61768,72094,83705,96711,111227,127371,145271,165056,186865,210838,237126,265881,
@@ -64017,7 +64017,7 @@ A064012 ,10,30,288,660,720,2146560,
A064013 ,1,35,215,225,398,2097,5205,7452,22359,98176,169653,
A064014 ,8281,183184,328329,528529,715716,60996100,82428241,98029801,1322413225,4049540496,106755106756,453288453289,538277538276,998002998001,20661152066116,29752082975209,2214532822145329,2802768328027684,7783702677837025,9998000299980001,
A064015 ,1,8,72,135,504,726,3431,5313,8614,10249,28721,45705,83832,115111,116057,235781,262844,349351,396815,530205,680229,2140452,3514448,5092315,5093695,9520080,12154006,12207991,12213847,13802199,13806381,13807119,29213727,115315480,
-A064016 ,0,23,2006,196308,19607514,1960399246,196036947608,19603648572758,1960364533634092,196036449326991586,19603644912113783634,1960364490766613788860,196036449073440195974090,
+A064016 ,0,23,2006,196308,19607514,1960399246,196036947608,19603648572758,1960364533634092,196036449326991586,19603644912113783634,1960364490766613788860,196036449073440195974090,19603644907302101080472556,1960364490729905106089642146,196036449072986990521291164848,
A064017 ,1,3,12,45,162,567,1944,6561,21870,72171,236196,767637,2480058,7971615,25509168,81310473,258280326,817887699,2582803260,8135830269,25569752274,80196041223,251048476872,784526490225,2447722649502,
A064018 ,1,32,3044,304192,30397486,3039650754,303963552392,30396356427242,3039635516365908,303963551173008414,30396355092886216366,3039635509283386211140,303963550927059804025910,30396355092702898919527444,3039635509270144893910357854,303963550927013509478708835152,
A064019 ,1,5,51,99,155,209,2369,2569,2882,5745,15143,21691,34573,36566,40516,41237,65304,82718,101638,112305,185701,238302,247221,254865,291399,439104,445794,483107,532645,538531,570020,690238,698561,772485,805013,
@@ -67249,7 +67249,7 @@ A067244 ,2,3,4,6,32,34,54,55,58,70,84,133,141,221,285,322,377,407,472,664,684,98
A067245 ,5,6,13,15,33,68,117,156,168,222,259,265,313,522,644,665,750,986,1065,1153,1178,1328,1351,1377,1447,1750,1812,1824,1976,2392,2482,3672,3885,4185,4460,4528,5450,5812,5824,6610,6804,7256,7306,8496,8930,9310,
A067246 ,1,2,6,8,11,14,22,30,70,86,102,116,130,140,154,186,238,286,390,406,422,454,459,646,830,869,1015,1070,1120,1518,1528,1710,1742,1870,2670,2871,3586,3654,4394,5070,5214,6102,6446,6692,7295,7943,8339,9204,10664,
A067247 ,1,2,4,6,10,16,25,39,63,99,158,253,402,639,1021,1633,2617,4153,6633,10460,16598,26146,41409,64733,102006,159165,
-A067248 ,7,9551,303027,440999,968819,5517973,27737957,93230839,46492090901,426836115943,732382677641,4895576080181,
+A067248 ,7,9551,303027,440999,968819,5517973,27737957,93230839,46492090901,426836115943,732382677641,4895576080181,77628540590583,3475456543097857,20396537622790811,
A067249 ,1,6,84,112,141,294,1188,1320,2508,4584,5406,8850,14270,17416,23320,31152,79035,117576,125576,132066,149877,160664,162514,164024,167970,170980,177744,184584,223286,1119636,1124592,1216644,1481800,1566920,1626716,
A067250 ,10,57,92,99,123,124,1677,2485,32578,33165,33220,451140,5954790,
A067251 ,1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,61,62,63,64,65,66,67,68,69,71,72,73,74,75,76,77,78,79,81,82,83,84,85,86,87,88,89,91,92,93,94,95,96,97,98,99,101,102,103,104,
@@ -68540,7 +68540,7 @@ A068535 ,1,2,4,8,16,32,35,64,128,256,297,512,1024,1093,2048,2186,2590,3279,3511,
A068536 ,88209,90288,125928,196020,368280,829521,1978020,2328480,5513508,8053155,19798020,86531940,197998020,554344560,556326540,1960396020,1979998020,5543944560,5925169800,8820988209,9028890288,12592925928,14011538112,19602196020,19799998020,
A068537 ,2,5,8,9,10,13,16,17,18,20,25,26,28,29,32,33,34,35,37,40,41,45,50,52,53,54,58,61,64,65,68,72,73,74,80,82,85,89,90,91,97,98,100,
A068538 ,111111111,111111111111111111,1111111111111111111111,111111111111111111111111111,111111111111111111111111111111111111,111111111111111111111111111111111111111111,
-A068539 ,5,25,120,580,2800,13500,65100,313940,1513820,7299760,35200020,169736780,818482260,
+A068539 ,1,5,25,120,580,2800,13500,65100,313940,1513820,7299760,35200020,169736780,818482260,3946776920,19031623000,
A068540 ,5,51,171,357,442,582,1071,1250,1292,1460,1746,5456,6435,6825,7248,8060,8140,8540,9348,9486,9516,9594,9632,9636,10476,10860,10950,12192,20910,23160,23680,25308,27468,28032,29340,31392,34000,37488,45220,
A068541 ,2,4,6,8,32,64,66,70,72,316,318,326,328,330,332,336,606,636,638,654,670,672,678,680,828,830,832,834,836,838,840,842,844,846,850,880,882,884,898,900,902,904,906,908,914,916,918,928,942,948,962,964,966,968,
A068542 ,3,1,370,123456790,411522633744855967078189300,137174211248285322359396433470507544581618655692729766803840877914951989026063100,
@@ -72215,7 +72215,7 @@ A072210 ,1,1,2,3,5,8,31,12,43,55,98,441,332,773,16,834,994,739,6341,3732,9083,28
A072211 ,1,1,2,2,4,1,6,2,3,1,10,1,12,1,1,2,16,1,18,1,1,1,22,1,5,1,3,1,28,1,30,2,1,1,1,1,36,1,1,1,40,1,42,1,1,1,46,1,7,1,1,1,52,1,1,1,1,1,58,1,60,1,1,2,1,1,66,1,1,1,70,1,72,1,1,1,1,1,78,1,3,1,82,1,1,1,1,1,88,1,1,1,1,1,
A072212 ,6,13,22,33,46,61,77,95,115,136,160,185,211,240,270,302,335,370,407,445,486,527,571,616,663,711,761,813,867,922,978,1037,1097,1159,1222,1287,1354,1422,1492,1563,1637,1711,1788,1866,1946,2027,2110,2195,2281,
A072213 ,1,1,5,30,231,1958,17977,173525,1741630,18004327,190569292,2056148051,22540654445,250438925115,2814570987591,31946390696157,365749566870782,4219388528587095,49005643635237875,572612058898037559,
-A072214 ,1,1,2,3,7,22,101,792,12310,451276,49995925,22540654445,60806135438329,1596675274490756791,758949605954969709105721,14362612091531863067120268402228,29498346711208035625096160181520548669694,23537552807178094028466621551669121053281242290608650,
+A072214 ,1,1,1,2,3,7,22,101,792,12310,451276,49995925,22540654445,60806135438329,1596675274490756791,758949605954969709105721,14362612091531863067120268402228,29498346711208035625096160181520548669694,23537552807178094028466621551669121053281242290608650,
A072215 ,4,5,7,15,176,476715857290,
A072216 ,5,89,187,1297,10911,150296,9008299,15002893,140669390,1005499526,10087799570,
A072217 ,2,24,23,21,55,64,96,96,98,109,149,149,188,186,201,197,236,232,
@@ -72242,7 +72242,7 @@ A072237 ,39916800,362880,371993326789901217467999448150835200000000,209227898880
A072238 ,720,87178291200,6402373705728000,8222838654177922817725562880000000,608281864034267560872252163321295376887552831379210240000000000,1124000727777607680000,230843697339241380472092742683027581083278564571807941132288000000000000,
A072239 ,11,12,15,27,192,299016608,
A072240 ,6,1307674368000,25852016738884976640000,403291461126605635584000000,263130836933693530167218012160000000,33452526613163807108170062053440751665152000000000,10333147966386144929666651337523200000000,
-A072241 ,1,1,1,2,3,6,18,76,512,6378,173682,12769602,3328423936,4338469000206,43848229368772905,5999189517441089061374,22578203777383772718280932410,5759108897879943749493986821813718586,
+A072241 ,1,1,1,1,2,3,6,18,76,512,6378,173682,12769602,3328423936,4338469000206,43848229368772905,5999189517441089061374,22578203777383772718280932410,5759108897879943749493986821813718586,313503492905074747917062873989282073311633745920,
A072242 ,5040,355687428096000,15511210043330985984000000,265252859812191058636308480000000,13763753091226345046315979581580902400000000,
A072243 ,1,1,2,8,32,142,668,3264,16444,84756,444793,2368800,12769602,69545358,382075868,2114965120,11784471548,66043042088,372022512608,2105220502772,11962163400706,68223286792200,390406746862530,2240962117491470,12899456450932840,
A072244 ,24,121645100408832000,51090942171709440000,620448401733239439360000,2658271574788448768043625811014615890319638528000000000,1405006117752879898543142606244511569936384000000000,
@@ -73166,7 +73166,7 @@ A073161 ,1,4,6,10,12,20,21,26,28,36,38,46,50,56,57,64,69,80,81,87,92,99,104,112,
A073162 ,1,3,17,37,9107,156335,679083,1068131,4883039,101691357,
A073163 ,0,3,68,259,5500628,1180641920,19503263760,46464766631,863653341852,306757978180563,
A073164 ,0,1,4,7,604,7552,28720,43501,176868,3016559,
-A073165 ,1,1,1,1,2,1,1,3,4,1,1,4,10,8,1,1,5,20,35,16,1,1,6,35,112,126,32,1,1,7,56,294,672,462,64,1,1,8,84,672,2772,4224,1716,128,1,1,9,120,1386,9504,28314,27456,6435,256,1,1,10,165,2640,28314,151008,306735,183040,
+A073165 ,1,1,1,1,2,1,1,3,4,1,1,4,10,8,1,1,5,20,35,16,1,1,6,35,112,126,32,1,1,7,56,294,672,462,64,1,1,8,84,672,2772,4224,1716,128,1,1,9,120,1386,9504,28314,27456,6435,256,1,1,10,165,2640,28314,151008,306735,183040,24310,512,1,
A073166 ,1,1,1,1,3,1,1,2,2,1,1,5,10,5,1,1,3,5,5,3,1,1,7,7,35,7,7,1,1,4,28,14,
A073167 ,4,6,12,36,158,871,5802,44996,396465,3905769,42492571,505638346,6530063762,90937897520,1358169957289,21652573590950,366977865386054,6588521300048437,124902980656633121,2493219952752601419,52268714816806571926,
A073168 ,1,5,9,9,9,15,16,17,18,17,25,19,21,27,31,26,21,22,21,35,38,31,35,31,34,33,37,39,49,49,33,52,49,47,39,43,47,47,48,48,41,49,48,60,59,59,49,52,58,58,63,71,75,65,65,67,71,79,75,81,84,77,65,69,72,72,67,69,61,65,65,
@@ -74979,7 +74979,7 @@ A074974 ,15,26,33,39,50,51,57,62,68,69,75,79,82,86,93,97,99,118,127,141,147,165,
A074975 ,24,42,66,96,104,108,114,140,156,174,176,180,222,224,228,270,282,288,336,352,354,392,396,400,444,448,464,516,532,534,560,572,576,594,644,650,666,702,704,708,714,720,740,756,760,774,780,800,810,822,828,870,
A074976 ,3,2,2,1,3,2,4,2,2,5,2,3,6,3,2,2,8,3,4,8,3,4,3,2,5,10,5,10,5,2,6,4,12,2,12,4,4,6,4,4,13,3,14,7,14,2,2,7,15,8,5,15,3,5,5,5,16,6,8,17,3,2,9,18,9,3,6,4,19,9,6,5,6,6,10,7,5,10,5,4,20,4,21,7,10,7,5,11,21,11,4,5,11,6,11,
A074977 ,1,10,55,161,209,551,649,1079,1189,3401,6049,6319,9701,12151,14279,
-A074978 ,6447,529271,569513,996733,
+A074978 ,6447,529271,569513,996733,53172153,837071903,53552588203,445839739269,6130987583999,
A074979 ,113,139,181,199,241,283,293,317,467,509,523,577,619,661,773,829,839,863,887,953,1021,1039,1069,1129,1237,1307,1327,1381,1459,1499,1583,1627,1637,1669,1699,1759,1789,1879,1913,1951,2003,2039,2089,2113,2143,
A074980 ,6,10,14,34,42,46,50,58,62,66,70,78,82,86,90,102,110,114,122,130,134,158,162,166,178,182,194,202,206,210,214,226,230,234,238,246,250,254,258,266,274,278,290,302,306,310,314,322,326,330,338,354,358,374,378,
A074981 ,6,14,34,42,50,58,62,66,70,78,82,86,90,102,110,114,130,134,158,178,182,202,206,210,226,230,238,246,254,258,266,274,278,290,302,306,310,314,322,326,330,358,374,378,390,394,398,402,410,418,422,426,
@@ -75064,7 +75064,7 @@ A075059 ,2,2,3,7,13,61,61,421,841,2521,2521,27721,27721,360361,360361,360361,720
A075060 ,10,40,146,427,1055,2510,5047,
A075061 ,2,3,4,7,8,9,13,14,15,16,61,62,63,64,65,61,62,63,64,65,66,421,422,423,424,425,426,427,841,842,843,844,845,846,847,848,2521,2522,2523,2524,2525,2526,2527,2528,2529,2521,2522,2523,2524,2525,2526,2527,2528,2529,
A075062 ,2,7,24,58,315,381,2968,6756,22725,25255,304986,332718,4684771,5045145,5405520,11531656,208288233,220540491,4423058830,4655851410,4888643991,5121436573,123147264516,128501493420,669278610325,696049754751,
-A075063 ,5,13,73,73,1801,1801,2521,2521,15121,15121,15121,15121,55441,55441,1108801,14414401,43243201,43243201,43243201,43243201,43243201,
+A075063 ,5,13,73,73,1801,1801,2521,2521,15121,15121,15121,15121,55441,55441,1108801,14414401,43243201,43243201,43243201,43243201,43243201,367567201,367567201,367567201,13967553601,13967553601,13967553601,13967553601,13967553601,13967553601,
A075064 ,9,25,91,841,6931,30031,510511,9699691,223092871,6469693231,601681470391,7420738134811,304250263527211,13082761331670031,614889782588491411,32589158477190044731,1922760350154212639071,
A075065 ,1,4,9,6,15,8,21,10,25,12,27,14,33,16,35,18,39,20,45,22,49,24,51,26,55,28,57,30,63,32,65,34,69,36,75,38,77,40,81,42,85,44,87,46,91,48,93,50,95,52,99,54,105,56,111,58,115,60,117,62,119,64,121,66,
A075066 ,9,4,15,6,21,8,25,10,27,12,33,14,35,16,39,18,45,20,49,22,51,24,55,26,57,28,63,30,65,32,69,34,75,36,77,38,81,40,85,42,87,44,91,46,93,48,95,50,99,
@@ -77218,7 +77218,7 @@ A077213 ,2,3,7,23,97,487,2927,20507,164057,1476523,14765237,162417611,1949011333
A077214 ,1,2,9,10,11,12,13,14,17,18,19,20,21,23,24,25,26,27,28,29,30,31,32,33,37,38,41,42,43,44,47,48,49,50,51,52,53,54,55,56,57,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,77,78,79,80,81,82,83,84,85,88,89,90,
A077215 ,1,1,4,3,16,4,32,9,8,64,5,256,16,7,8,128,32,4,256,8,8,16,32,64,512,8,17,16,27,32,2048,256,64,23,1024,25,64,128,8,64,64,9,256,64,8,9,4096,1024,64,4,16,16,16,256,2048,16,64,27,128,16,47,2048,8192,8,8,8,32768,128,
A077216 ,2,5,3,19,7,13,23,47,31,89,139,113,199,293,631,317,1069,509,2503,1129,1759,2039,887,1951,4027,3967,2477,2971,3271,6917,4831,5591,10799,5119,14107,9973,1327,39461,16381,20809,11743,15683,61169,52391,33247,45439,
-A077217 ,2,5,17,29,35,41,101,137,
+A077217 ,2,5,17,29,41,101,107,137,149,179,197,269,281,457,461,499,521,569,593,617,641,673,727,809,821,827,857,881,1049,1061,1229,1277,1289,1301,1321,1451,1453,1481,1483,1619,1697,1721,1753,1777,1861,1873,1877,1949,1997,2027,
A077218 ,0,2,2,7,3,8,3,7,14,3,15,8,3,8,15,14,4,16,8,5,13,11,14,21,10,3,9,5,10,36,12,16,3,26,4,16,17,8,16,15,5,26,7,9,4,33,30,12,4,10,14,6,29,20,14,15,5,17,10,3,28,40,9,5,9,42,16,27,4,14,13,22,17,18,8,19,22,11,23,27,5,
A077219 ,1,1,1,1,2,2,2,3,3,3,4,4,5,5,5,6,6,6,7,7,7,8,9,8,9,9,10,10,11,11,12,12,12,12,12,13,14,14,14,14,15,15,16,16,16,16,17,17,18,18,18,19,20,19,20,20,20,21,22,22,23,22,23,23,23,24,25,25,25,25,26,26,27,27,27,27,28,28,
A077220 ,1,2,4,6,9,12,3,7,8,13,15,21,24,31,5,10,11,17,19,26,29,16,20,25,30,36,42,49,56,22,14,41,37,18,27,28,38,40,51,54,66,39,52,53,67,69,84,87,33,45,46,32,23,43,35,70,50,55,65,71,34,44,47,58,62,74,79,57,48,72,64,89,
@@ -77655,7 +77655,7 @@ A077650 ,1,3,4,7,6,1,8,1,1,1,1,2,1,2,2,3,1,3,2,4,3,3,2,6,3,4,4,5,3,7,3,6,4,5,4,9
A077651 ,1,1,2,2,4,2,6,4,6,4,1,4,1,6,8,8,1,6,1,8,1,1,2,8,2,1,1,1,2,8,3,1,2,1,2,1,3,1,2,1,4,1,4,2,2,2,4,1,4,2,3,2,5,1,4,2,3,2,5,1,6,3,3,3,4,2,6,3,4,2,7,2,7,3,4,3,6,2,7,3,5,4,8,2,6,4,5,4,8,2,7,4,6,4,7,3,9,4,6,4,1,3,1,4,4,
A077652 ,2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,929,1021,1031,1051,1061,1091,1151,1171,1181,1201,1231,1291,1301,1321,1361,1381,1451,1471,1481,1511,1531,1571,1601,1621,1721,1741,1801,1811,
A077653 ,1,2,2,1,3,0,4,1,3,2,2,3,1,4,0,5,1,4,2,3,3,2,4,1,5,0,6,1,5,2,4,3,3,4,2,5,1,6,0,7,1,6,2,5,3,4,4,3,5,2,6,1,7,0,8,1,7,2,6,3,5,4,4,5,3,6,2,7,1,8,0,9,1,8,2,7,3,6,4,5,5,4,6,3,7,2,8,1,9,0,10,1,9,2,8,3,7,4,6,5,5,6,4,7,
-A077654 ,4,10,12,16,22,24,25,27,28,32,34,38,40,42,45,46,49,52,55,57,58,60,62,64,66,70,72,76,77,80,82,84,85,87,88,91,92,93,94,100,102,104,106,108,110,112,115,117,118,121,122,123,124,126,129,130,
+A077654 ,4,10,12,16,22,24,25,27,28,32,34,38,40,42,45,46,49,52,55,57,58,60,62,64,66,70,72,76,77,80,82,84,85,87,88,91,92,93,94,100,102,104,106,108,110,112,115,117,118,121,122,123,124,126,129,130,132,133,136,142,
A077655 ,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0,0,
A077656 ,1,3,4,5,6,7,8,10,11,12,13,15,16,17,18,19,20,22,23,24,26,28,29,30,31,32,35,36,37,39,40,41,42,43,45,46,47,48,49,50,51,52,53,54,55,56,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,76,77,78,79,80,81,82,83,
A077657 ,1,2,33,603,602,2522,211673,3405123,3405122,49799889,202536181,3195380868,5208143601,85843948321,97524222465,
@@ -79156,7 +79156,7 @@ A079151 ,2,3,5,7,11,13,19,23,29,31,43,47,53,59,67,71,79,83,103,107,131,139,149,1
A079152 ,2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,97,101,103,107,109,113,131,137,139,149,157,163,167,173,179,181,191,193,211,223,227,229,239,241,257,263,269,277,281,283,293,311,313,317,331,
A079153 ,2,3,5,7,11,13,19,29,43,67,173,283,317,653,787,907,1867,2083,2693,2803,3413,3643,3677,4253,4363,4723,5443,5717,6197,6547,6653,8563,8573,9067,9187,9403,9643,10733,11443,11587,12163,12917,13997,14107,14683,15187,
A079154 ,3,4,12,30,33,1406,
-A079155 ,4,15,85,619,4800,39266,332276,2880818,
+A079155 ,4,15,85,619,4800,39266,332276,2880818,25423985,227527467,
A079156 ,10,67,396,2201,11870,62571,324896,1665349,8457890,42605267,213305636,1061939193,5263752278,25984214383,127848694424,627084275649,3067923454498,
A079157 ,5,50,529,3870,40150,185014,1191698,7080332,
A079158 ,5,40,399,2472,17436,98400,601626,3238694,
@@ -82217,7 +82217,7 @@ A082212 ,5,136,2491,949777,332986830284,174484936587602528213815,968636210166473
A082213 ,1,3,4,181,594448268301656413948075911105052760867948344134387820089804440720816962,
A082214 ,13,34,4181,181594448268301656413948075911105052760867948344134387820089804440720816962,
A082215 ,1,121,1213121,121312141213121,1213121412131215121312141213121,121312141213121512131214121312161213121412131215121312141213121,1213121412131215121312141213121612131214121312151213121412131217121312141213121512131214121312161213121412131215121312141213121,
-A082216 ,1,2,3,4,5,6,7,8,9,101,11,121,131,141,151,161,171,181,191,202,212,22,232,242,252,262,272,282,292,303,313,323,33,343,353,363,373,383,393,404,414,424,434,44,454,464,474,484,494,505,
+A082216 ,0,1,2,3,4,5,6,7,8,9,101,11,121,131,141,151,161,171,181,191,202,212,22,232,242,252,262,272,282,292,303,313,323,33,343,353,363,373,383,393,404,414,424,434,44,454,464,474,484,494,505,
A082217 ,1,2,3,4,5,6,7,8,9,10801,11711,12621,13531,14441,15351,16261,17171,1881,0,208802,2139312,227722,2329232,246642,2519152,265562,
A082218 ,1,3,5,6,7,2,10,12,8,4,9,14,13,16,19,25,15,37,21,23,11,20,17,22,29,26,35,24,28,36,18,32,38,44,40,48,31,56,33,68,43,50,39,34,41,27,47,61,53,57,45,75,85,93,55,30,49,65,63,72,67,88,69,62,73,51,81,83,80,70,128,42,
A082219 ,1,3,2,10,19,25,24,28,41,27,51,81,78,86,124,120,147,123,188,142,192,116,258,250,314,254,320,392,470,404,453,377,490,612,533,445,718,708,812,602,784,726,791,771,928,1002,1032,1158,996,972,1149,1023,1365,1239,
@@ -82789,13 +82789,13 @@ A082784 ,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1
A082785 ,0,0,0,0,0,1,1,1,2,0,3,2,2,1,2,2,5,1,2,2,3,2,4,1,2,3,4,4,2,1,1,4,3,2,2,1,3,5,1,3,2,2,2,3,2,3,3,2,1,2,4,5,2,3,1,4,3,3,1,1,1,5,2,2,2,3,2,5,1,1,2,3,2,2,2,3,3,3,1,3,2,4,2,1,2,3,2,4,2,1,0,4,3,3,1,3,1,3,1,1,1,2,2,3,1,
A082786 ,0,1,0,0,1,0,2,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,
A082787 ,2,60,6720,1663200,726485760,494010316800,482718652416000,641171050071552000,1111363153457356800000,2436552577639909048320000,6591982246414881207091200000,21572261901392698750205952000000,83992431415453295380032651264000000,383725422380885198036206312488960000000,
-A082788 ,1,259,1260,2071,2834,7574,7749,9252,12800,18720,28236,30039,32724,42120,45395,45877,68124,102656,135813,246543,264440,341288,389455,423163,480624,523775,936471,937248,
+A082788 ,1,259,1260,2071,2834,7574,7749,9252,12800,18720,28236,30039,32724,42120,45395,45877,68124,102656,135813,246543,264440,341288,389455,423163,480624,523775,936471,937248,1055954,1182104,1295749,1333626,1366632,1379196,1458270,1483118,
A082789 ,1,2,5,16,56,282,1865,17100,207697,3180571,
A082790 ,0,0,0,1,1,5,19,153,1615,25180,479238,10695820,
A082791 ,2,1,7,5,4,4,3,3,3,2,2,2,2,2,14,13,12,12,11,1,1,1,1,1,1,1,1,1,1,7,7,7,7,6,6,6,6,6,6,5,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
A082792 ,3,30,3,32,30,30,35,32,36,30,33,36,39,308,30,32,34,36,38,300,315,308,322,312,300,312,324,308,319,30,31,32,33,34,35,36,37,38,39,320,328,336,301,308,315,322,329,336,343,300,306,312,318,324,330,336,342,348,354,
A082793 ,1,1,1,2,1,2,4,2,2,4,7,4,4,4,7,13,7,8,8,7,13,24,13,14,16,14,13,24,44,24,26,28,28,26,24,44,
-A082794 ,4,4,42,4,40,42,42,40,45,40,44,48,403,42,45,48,408,414,418,40,42,44,46,48,400,416,405,420,406,420,403,416,429,408,420,432,407,418,429,40,41,42,43,44,45,46,47,48,49,400,408,416,424,432,440,448,
+A082794 ,4,4,42,4,40,42,42,40,45,40,44,48,403,42,45,48,408,414,418,40,42,44,46,48,400,416,405,420,406,420,403,416,429,408,420,432,407,418,429,40,41,42,43,44,45,46,47,48,49,400,408,416,424,432,440,448,456,406,413,
A082795 ,5,50,51,52,5,54,56,56,54,50,55,504,52,56,510,512,51,54,57,500,504,506,506,504,50,52,54,56,58,510,527,512,528,510,525,504,518,532,507,520,533,504,516,528,540,506,517,528,539,50,51,52,53,54,
A082796 ,6,6,6,60,60,6,63,64,63,60,66,60,65,602,60,64,68,612,608,60,63,66,69,600,600,624,621,616,609,60,62,64,66,68,630,612,629,608,624,600,615,630,602,616,630,644,611,624,637,600,612,624,636,648,605,
A082797 ,7,70,72,72,70,72,7,72,72,70,77,72,78,70,75,704,714,72,76,700,714,704,713,72,75,78,702,700,725,720,713,704,726,714,70,72,74,76,78,720,738,714,731,704,720,736,705,720,735,700,714,728,742,702,715,
@@ -83548,7 +83548,7 @@ A083543 ,3,12,28,42,2,6,120,195,234,6,21,2,84,1,744,558,78,780,210,336,72,6,10,
A083544 ,1,2,3,3,4,5,6,6,7,8,9,9,10,11,12,12,12,13,14,14,15,16,17,17,
A083545 ,1,3,15,19,95,104,125,164,194,255,259,341,491,495,504,512,513,584,591,629,679,755,775,975,1024,1147,1247,1254,1260,1313,1358,1463,1469,1538,1615,1728,1919,1962,1970,2047,2071,2090,2204,2299,2321,2345,2404,2625,
A083546 ,1,2,8,12,48,48,60,80,96,128,144,180,280,240,240,288,288,288,336,288,384,360,480,480,640,720,672,600,576,720,720,720,672,864,960,864,960,1080,1008,1408,1296,960,1008,1320,1260,1056,1440,1200,1728,1440,1296,
-A083547 ,1,12,24,36,36,60,60,72,80,96,120,120,120,144,144,168,180,240,264,360,360,432,480,504,480,480,720,720,
+A083547 ,1,12,24,36,36,60,60,72,80,96,120,120,120,144,144,168,180,240,264,360,360,432,480,504,480,480,720,720,720,720,840,840,864,840,840,840,840,960,900,960,960,1080,1260,1224,1320,1320,1440,1440,1320,1440,1440,1728,
A083548 ,0,1,2,2,4,4,4,12,6,6,8,8,8,56,56,8,12,12,12,36,36,12,16,80,70,126,144,16,22,22,16,208,234,198,264,24,20,60,120,24,30,30,24,168,168,24,32,224,210,570,532,28,36,180,480,672,210,30,44,44,32,864,864,544,782,46,36,
A083549 ,0,1,2,2,4,4,4,12,2,6,8,8,8,56,56,8,12,12,12,12,12,12,16,80,70,126,144,16,22,22,16,208,234,198,264,24,20,12,40,24,30,30,24,56,56,24,32,224,210,570,532,28,36,60,480,672,70,30,44,44,32,864,864,544,782,46,36,900,
A083550 ,2,4,8,8,8,8,8,24,12,12,24,8,8,24,36,12,12,24,8,12,24,24,48,32,8,8,8,8,56,56,24,12,20,20,12,36,24,24,36,12,20,20,8,8,24,144,48,8,8,24,12,20,60,36,36,12,12,24,8,20,140,56,8,8,56,84,60,20,8,24,48,48,36,24,24,48,
@@ -84320,7 +84320,7 @@ A084315 ,1,3,4,13,36,19,120,33,46,11,78,37,560,239,496,1905,52,397,3250,221,778,
A084316 ,1,3,20,11,24,5,6,39,98,29,120,23,64,13,104,15,1716,323,284,499,62,1099,1264,215,1274,51,512,447,1768,209,1332,31,32,373,34,1475,258,835,2300,519,5780,419,5374,1275,6974,1655,6626,479,10240,10549,3008,883,13938,
A084317 ,0,2,3,2,5,23,7,2,3,25,11,23,13,27,35,2,17,23,19,25,37,211,23,23,5,213,3,27,29,235,31,2,311,217,57,23,37,219,313,25,41,237,43,211,35,223,47,23,7,25,317,213,53,23,511,27,319,229,59,235,61,231,37,2,513,2311,67,
A084318 ,0,2,3,2,5,23,7,2,3,5,11,23,13,3,1129,2,17,23,19,5,37,211,23,23,5,3251,3,3,29,547,31,2,311,31397,1129,23,37,373,313,5,41,379,43,211,1129,223,47,23,7,5,317,3251,53,23,773,3,1129,229,59,547,61,31237,37,2,1129,2311,
-A084319 ,91,713,2331,3737,37101,383149,1329473,10912197,328312853,1129846623,3735159117,31245053039,173977184859,3293176308321,319269241788861,371325123869195203,1278647733810375857,1665622037676698019,
+A084319 ,91,713,2331,3737,37101,383149,1329473,10912197,328312853,1129846623,3735159117,31245053039,173977184859,3293176308321,319269241788861,371325123869195203,1278647733810375857,1665622037676698019,31742715741254857303,56627509560552923867,
A084320 ,1,1,2,2,3,3,3,3,3,4,3,4,4,4,4,4,4,4,5,4,4,5,5,4,5,5,4,5,5,5,5,5,5,5,6,5,5,5,6,5,5,6,5,6,5,6,5,6,6,5,6,6,6,5,6,6,6,6,6,6,6,5,6,7,6,6,6,6,6,6,6,6,7,6,6,6,7,6,6,7,6,6,7,6,7,6,7,6,6,7,7,6,7,6,7,6,7,7,6,7,7,6,7,7,6,
A084321 ,1,3,5,10,19,35,64,139,256,536,1061,2095,4169,8282,16517,32903,65646,131205,262579,525083,1048893,2098826,4195521,8390583,16782032,33569609,67118347,134229613,268453180,536890474,1073764782,2147523518,
A084322 ,2,23,547,2357,4359293547691,325798243129564339,3947306373286437248759663633906484193454376823,
@@ -86622,7 +86622,7 @@ A086617 ,1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,13,13,5,1,1,6,21,33,21,6,1,1,7,31,69,
A086618 ,1,2,7,33,183,1118,7281,49626,349999,2535078,18758265,141254655,1079364105,8350678170,65298467487,515349097713,4100346740511,32858696386766,265001681344569,
A086619 ,1,2,10,150,7650,1438200,1051324200,3101406390000,37945707181650000,1966422437567466300000,438887790263120370963300000,427664112802721593716655529100000,
A086620 ,1,1,1,1,3,1,1,5,5,1,1,7,14,7,1,1,9,28,28,9,1,1,11,47,79,47,11,1,1,13,71,175,175,71,13,1,1,15,100,331,504,331,100,15,1,1,17,134,562,1196,1196,562,134,17,1,1,19,173,883,2464,3514,2464,883,173,19,1,1,21,217,
-A086621 ,1,3,14,79,504,3514,26172,204831,1664696,13930840,119312544,1041227642,9228614836,82867255956,752405060536,6897376441167,63760133568096,593763928313128,
+A086621 ,1,3,14,79,504,3514,26172,204831,1664696,13930840,119312544,1041227642,9228614836,82867255956,752405060536,6897376441167,63760133568096,593763928313128,5565678569009328,52475976165495960,497376657383374560,4736680863568248480,45304174896889357440,
A086622 ,1,2,5,12,30,76,197,520,1398,3820,10594,29768,84620,243000,704045,2055760,6043750,17875020,53148310,158773320,476311940,1434313960,4333867170,13135533552,39924668220,121661345656,371612931492,
A086623 ,1,1,1,1,1,1,1,2,2,1,1,3,4,3,1,1,4,8,8,4,1,1,5,14,19,14,5,1,1,6,22,40,40,22,6,1,1,7,32,76,100,76,32,7,1,1,8,44,132,222,222,132,44,8,1,1,9,58,213,448,570,448,213,58,9,1,1,10,74,324,834,1316,1316,834,324,74,10,1,1,
A086624 ,1,1,4,19,100,570,3442,21685,141112,941990,6419174,44493000,312818326,2226155632,16008452202,116167346499,849724397580,6259403310366,46399703925202,345894094030552,
@@ -87317,7 +87317,7 @@ A087312 ,1,3,5,9,17,33,131,143055667,
A087313 ,2,3,255,1046,230584301136334848,7065193045869401568154708608,
A087314 ,2,4,20,1154,3907502,609516468354242,108233912076063807870514781000,205688069665244291374160325606433848956971528595913979304345602,381367496233593231179533022742555015402552706280473714446093438118953849830784189071820024395733993533363857256493600698,
A087315 ,1,4,72,21600,190512000,580909190400000,428616352408083840000000,859278392084450410309036800000000000,2097197194438629126172451944256706311040000000000000,
-A087316 ,4,17,84,545,7824,281771,51540600,3347558057,1146374959980,288113965730819,529172633067826888,283453407513524913023,4122282265785671687518812,1586581830624893452605127040309,
+A087316 ,4,17,84,545,7824,281771,51540600,3347558057,1146374959980,288113965730819,529172633067826888,283453407513524913023,4122282265785671687518812,1586581830624893452605127040309,412109111737176949907195758658736,
A087317 ,2,3,2,3,2,5,14,3,2,3,2,5,21,3,2,3,2,5,8,3,2,7,10,5,10,3,2,3,2,7,15,5,6,
A087318 ,0,2,2,0,2,217,2,0,0,10001,2,0,2,15,226,0,2,325,2,0,22,485,2,0,0,456977,0,0,2,27001,2,0,34,35,1226,0,2,39,1522,0,2,130691233,2,45,0,2117,2,0,0,0,2602,0,2,0,0,0,58,3365,2,0,2,3845,0,0,4226,287497,2,0,4762,24010001,
A087319 ,0,2,2,0,2,217,2,65,0,10001,2,0,2,15,226,0,2,325,2,0,22,485,2,110075314177,0,27,0,0,2,27001,2,1025,34,35,1226,0,2,39,1522,64001,2,130691233,2,45,0,2117,2,0,0,0,2602,0,2,1338925209985,0,
@@ -87423,7 +87423,7 @@ A087418 ,2,3,3,3,5,5,5,5,7,7,7,7,7,11,11,11,13,13,19,23,23,29,31,37,37,41,41,41,
A087419 ,1,1,2,4,1,2,3,4,2,3,5,7,10,6,7,12,8,10,9,16,23,8,13,30,40,13,20,25,27,33,27,29,20,8,29,66,37,52,8,44,71,99,47,79,59,105,104,60,106,12,13,50,121,173,167,3,49,34,7,42,42,182,107,53,157,197,314,8,335,211,273,229,
A087420 ,2,4,14,14,34,28,62,46,98,68,142,94,194,124,254,158,322,196,398,238,482,284,574,334,674,388,782,446,898,508,1022,574,1154,644,1294,718,1442,796,1598,878,1762,964,1934,1054,2114,1148,2302,1246,2498,1348,2702,
A087421 ,2,2,2,7,29,127,727,5051,40343,362897,3628811,39916801,479001629,6227020867,87178291219,1307674368043,20922789888023,355687428096031,6402373705728037,121645100408832089,2432902008176640029,
-A087422 ,1,2,8,55,567,7958,142396,3104160,
+A087422 ,1,2,8,55,567,7958,142396,3104160,79813513,
A087423 ,32,768,20672,565248,15491072,424685568,11643256832,319215894528,8751751626752,239941585993728,6578336360824832,180354352643309568,4944668491903926272,135565048129674805248,3716706651755063017472,101898745479045492768768,
A087424 ,567,239841,114082668,55125843489,26697877691247,12934267027240356,6266540498895923463,3036106030479071781249,1470978970343729016987852,712682440446248640284336721,345291321126117622870522555983,167292036479044881831300837903684,81052212349412217472309893818152407,
A087425 ,23105,459119455,9758296035305,208416652653910655,4452963734477926435505,95143212432467064852443605,2032859482921447476046969568705,43434715031065603778576465510557055,
@@ -87915,7 +87915,7 @@ A087910 ,1,2,2,5,8,5,5,13,9,10,10,12,12,12,12,22,17,18,18,21,22,21,21,27,25,26,2
A087911 ,2,3,5,17,191,257,1009,4561,4591,21601,57601,54121,86677,176401,415801,291721,950041,1259701,3049201,1670761,6098401,3880801,5654881,13759201,18618601,14414401,18960481,15135121,31600801,45405361,35814241,
A087912 ,1,3,14,86,648,5752,58576,671568,8546432,119401856,1815177984,29808908032,525586164736,9898343691264,198227905206272,4204989697906688,94163381359509504,2219240984918720512,54898699229094412288,1422015190821016633344,38484192401958599131136,
A087913 ,0,1,1,3,1,5,1,6,7,7,1,10,1,13,13,12,1,15,1,15,19,19,1,20,23,23,26,26,1,26,1,24,31,31,31,30,1,37,37,30,1,39,1,39,39,43,1,40,47,49,47,51,1,52,53,52,53,53,1,52,1,61,62,48,61,65,1,65,67,69,1,60,1,73,74,74,73,
-A087914 ,8,48,224,960,1215,3968,16128,65024,261120,1046528,4190208,
+A087914 ,8,48,224,960,1215,3968,16128,65024,261120,1046528,4190208,16769024,67092480,268402688,1073676288,4294836224,
A087915 ,0,2,4,8,10,14,20,22,28,32,34,38,40,50,52,62,64,68,74,80,82,88,94,98,104,110,112,118,124,130,134,140,152,154,164,172,178,182,188,190,208,214,218,220,230,232,238,242,244,248,250,260,272,280,284,292,298,302,
A087916 ,0,0,0,0,1,3,6,7,9,12,16,18,21,27,30,30,34,36,42,46,48,48,51,63,60,64,81,75,76,87,87,90,102,105,97,117,114,105,144,129,126,159,141,145,177,162,160,195,186,153,207,201,171,237,210,187,255,234,222,279,
A087917 ,0,0,1,1,1,2,1,1,2,3,2,3,3,2,4,6,5,5,6,5,6,8,8,9,9,8,10,12,12,14,14,10,14,19,14,18,20,14,19,25,21,20,27,22,23,32,26,27,31,29,31,36,35,35,39,34,38,47,40,42,47,40,43,60,53,44,60,50,48,68,62,54,64,65,58,75,
@@ -96118,7 +96118,7 @@ A096113 ,1,2,3,6,4,8,12,18,24,36,48,72,144,5,10,15,16,20,30,32,40,54,60,64,80,90
A096114 ,1,2,3,5,4,6,10,11,9,8,7,12,19,20,21,23,22,18,16,17,15,14,13,24,37,38,39,41,40,42,46,47,45,44,43,36,31,32,33,35,34,30,28,29,27,26,25,48,73,74,75,77,76,78,82,83,81,80,79,84,91,92,93,95,94,90,88,89,87,86,85,72,
A096115 ,1,2,2,3,6,6,3,4,12,24,24,12,8,8,4,5,20,40,40,60,120,120,60,20,15,30,30,15,10,10,5,6,30,60,60,90,180,180,90,120,360,720,720,360,240,240,120,30,24,48,48,72,144,144,72,24,18,36,36,18,12,12,6,7,42,84,84,126,
A096116 ,1,2,3,5,4,9,7,6,5,11,12,14,9,10,8,7,6,13,14,16,15,20,18,17,11,12,13,15,10,11,9,8,7,15,16,18,17,22,20,19,18,24,25,27,22,23,21,20,13,14,15,17,16,21,19,18,12,13,14,16,11,12,10,9,8,17,18,20,19,24,22,21,20,26,
-A096117 ,0,0,0,0,1,1365,290745,35804384,3431889000,288982989000,22716104811840,1724903317684800,129165517275377154,9664573656742964960,728813888470620552600,55713446610261097382400,
+A096117 ,0,0,0,0,1,1365,290745,35804384,3431889000,288982989000,22716104811840,1724903317684800,129165517275377154,9664573656742964960,728813888470620552600,55713446610261097382400,4334305420045397178746260,344080024970397555374419968,27923503603736889921687649020,
A096118 ,1,1,2,3,4,7,9,10,11,21,30,37,41,44,46,47,48,95,141,185,226,263,293,314,
A096119 ,1,2,4,11,48,362,5030,133924,6977521,719087781,147394353130,60255915944715,49197429536084417,80280819225274033666,261914438169525117048056,
A096120 ,1,1,2,3,4,5,6,8,11,12,13,15,18,22,27,33,41,42,43,45,48,52,57,63,71,82,
@@ -96151,7 +96151,7 @@ A096146 ,2,5,7,19,71,97,3691,191861,138907099,708158977,26947261171,
A096147 ,3,11,41,571,2131,110771,1542841,15558008491,808717138331,1663476485027525263506023431291963826940251,33648911495192637123958375850447995878147331088460770783226682531,
A096148 ,2,3,5,7,23,37,53,73,223,233,337,523,733,773,5233,33377,72733,272333,572333,5222333,
A096149 ,2,3,5,7,9,11,13,15,19,35,39,45,51,59,213,607,1315,1435,3901,4921,5255,
-A096150 ,0,0,0,0,0,105,116175,37007656,7032842901,1016662746825,125217059384890,13979620699390500,1468384747758433362,148610523724144786304,14725179052834536611325,
+A096150 ,0,0,0,0,0,105,116175,37007656,7032842901,1016662746825,125217059384890,13979620699390500,1468384747758433362,148610523724144786304,14725179052834536611325,1444367897584925254381440,141356080305700826710780155,13881663444819892480039097856,
A096151 ,7,7,6,0,2,7,1,4,0,6,4,8,6,8,1,8,2,6,9,5,3,0,2,3,2,8,3,3,2,1,3,8,8,6,6,6,4,2,3,2,3,2,2,4,0,5,9,2,3,3,7,6,1,0,3,1,5,0,6,1,9,2,2,6,9,0,3,2,1,5,9,3,0,6,1,4,0,6,9,5,3,1,9,4,3,4,8,9,5,5,3,2,3,8,3,3,0,3,3,2,3,8,5,8,0,
A096152 ,8,0,4,8,2,5,7,5,9,6,5,9,6,4,0,3,3,6,8,8,8,5,1,8,8,4,2,2,7,9,3,2,2,8,7,8,3,5,1,0,0,4,5,4,0,6,3,7,7,9,5,5,3,0,1,7,1,7,3,8,0,0,9,2,5,8,8,1,9,1,3,8,1,0,0,3,4,6,5,5,6,7,1,4,3,3,7,8,9,5,6,0,3,4,1,3,4,3,9,5,3,9,7,1,0,
A096153 ,2,3,4,5,9,6,7,25,10,8,11,49,14,27,12,13,121,15,125,20,16,17,169,21,343,28,81,18,19,289,22,1331,44,625,50,24,23,361,26,2197,45,2401,75,40,30,29,529,33,4913,52,14641,98,56,42,32,31,841,34,6859,63,28561,147,88,66,
@@ -96225,7 +96225,7 @@ A096220 ,1,3,6,12,20,33,48,66,87,
A096221 ,1,2,3,5,7,10,13,15,18,
A096222 ,1,3,9,30,100,360,1296,4896,18496,71808,278784,1098240,4326400,17172480,68161536,271589376,1082146816,4320165888,17247043584,68920934400,275415040000,1101122764800,4402342526976,17605073043456,70403108110336,
A096223 ,1,1,3,4,11,8,29,26,52,49,138,79,271,198,337,389,914,477,1596,993,1881,1912,4507,2222,6485,5080,8682,7384,18459,6780,28628,19598,31098,29444,53198,30470,99132,65771,104464,80422,215307,81792,313064,195091,272503,
-A096224 ,0,0,0,0,0,15,54257,30258935,8403710364,1624745199910,253717024819170,34644709397517912,4336461198140896396,512755474242717445740,58441126001104710458595,
+A096224 ,0,0,0,0,0,15,54257,30258935,8403710364,1624745199910,253717024819170,34644709397517912,4336461198140896396,512755474242717445740,58441126001104710458595,6511044113057606391228960,716247426054164600104429648,78368395883181612191026677504,
A096225 ,1,2,3,7,71,6653,25469,15750503,
A096226 ,2,2,3,1,5,3,7,1,1,5,11,1,13,7,5,1,17,1,19,1,7,11,23,1,1,13,1,1,29,5,31,1,11,17,13,1,37,19,13,1,41,7,43,1,1,23,47,1,1,1,17,1,53,1,21,1,19,29,59,1,61,31,1,1,13,11,67,1,23,13,71,1,73,37,1,1,31,13,79,1,1,41,83,1,
A096227 ,2,8,16,44,96,268,648,1832,4784,13456,36832,102944,289216,804928,2292608,6365312,18257664,50626816,145731072,403833344,
@@ -96489,7 +96489,7 @@ A096484 ,1,10,105,1054,10540,105409,1054092,10540925,105409255,1054092553,105409
A096485 ,2,6,2,24,2,622,2,2396,2,21912,2,527718,2,168484,2,13171730,2,359947864,2,52090778,2,16658818532,2,134257065348,2,
A096486 ,8,170,2242,2132,1294,976846,216566,9904144,25617930,408928520,25346031262,137031675878,
A096487 ,2,6,20,66,210,666,2108,6666,21080,66666,210818,666666,2108184,6666666,21081850,66666666,210818510,666666666,2108185106,6666666666,21081851066,66666666666,210818510676,666666666666,2108185106778,
-A096488 ,2,3,2,8,2,37,2,76,2,217,2,870,2,583,2,5034,2,28494,2,10058,2,
+A096488 ,2,3,2,8,2,37,2,76,2,217,2,870,2,583,2,5034,2,28494,2,10058,2,187966,2,383291,2,340992,2,
A096489 ,1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
A096490 ,60,120,168,180,240,252,300,336,360,420,480,504,540,600,660,672,720,756,780,792,840,900,924,936,960,1008,1020,1080,1140,1176,1200,1260,1320,1344,1380,1440,1500,1512,1560,1584,1620,1680,1740,1764,1800,1848,1860,
A096491 ,1,2,2,2,4,4,4,4,3,6,6,6,6,6,6,4,8,8,8,8,8,8,8,8,5,10,10,10,10,10,10,10,10,10,10,6,12,12,12,12,12,12,12,12,12,12,12,12,7,14,14,14,14,14,14,14,14,14,14,14,14,14,14,8,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,
@@ -98450,8 +98450,8 @@ A098445 ,1,4,4,0,0,1,0,1,0,0,0,0,4,0,0,4,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,4
A098446 ,1,1,1,1,2,1,1,3,4,1,1,4,9,9,1,1,5,16,30,24,1,1,6,25,70,115,77,1,1,7,36,135,344,510,295,1,1,8,49,231,805,1908,2602,1329,1,1,9,64,364,1616,5325,11904,15133,6934,1,1,10,81,540,2919,12381,39001,83028,99367,41351,1,
A098447 ,1,1,1,1,2,1,1,3,4,1,1,4,9,9,1,1,5,16,32,24,1,1,6,25,78,150,79,1,1,7,36,155,532,1018,340,1,1,8,49,271,1395,5802,10996,2090,1,1,9,64,434,3036,21343,116658,212434,20613,1,1,10,81,652,5824,60209,661325,5072504,
A098448 ,1,2,4,9,24,79,340,2090,20613,374593,14797043,1558788465,568317523370,1002992052280356,13173490079341336160,2227644152149802108130325,9740856579902962818887540217002,
-A098449 ,4,10,106,1003,10001,100001,1000001,10000001,100000001,1000000006,10000000003,100000000007,1000000000007,10000000000015,100000000000013,1000000000000003,10000000000000003,100000000000000015,
-A098450 ,9,95,998,9998,99998,999997,9999998,99999997,999999991,9999999997,99999999997,999999999997,9999999999989,99999999999997,999999999999998,9999999999999994,99999999999999989,999999999999999993,
+A098449 ,4,10,106,1003,10001,100001,1000001,10000001,100000001,1000000006,10000000003,100000000007,1000000000007,10000000000015,100000000000013,1000000000000003,10000000000000003,100000000000000015,1000000000000000007,10000000000000000001,
+A098450 ,9,95,998,9998,99998,999997,9999998,99999997,999999991,9999999997,99999999997,999999999997,9999999999989,99999999999997,999999999999998,9999999999999994,99999999999999989,999999999999999993,9999999999999999991,99999999999999999983,
A098451 ,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,23,43,
A098452 ,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,21,22,43,
A098453 ,1,2,12,56,304,1632,9024,50304,283392,1607168,9167872,52537344,302239744,1744412672,10096263168,58576306176,340566147072,1983765676032,11574393962496,67631502065664,395710949228544,2318088492023808,13594307705438208,79802741538422784,468895276304695296,
@@ -101751,7 +101751,7 @@ A101746 ,7,103,2503,88903,4322503,2473107965928318342544472044975303,
A101747 ,3,4,5,6,7,19,40,56,93,
A101748 ,1,1,1,3,1,7,7,7,8,4,8,9,8,5,6,2,2,6,0,2,6,8,4,1,0,0,7,9,3,2,9,8,8,8,4,3,1,7,1,2,4,6,6,7,5,0,7,1,8,9,6,8,3,6,3,3,8,4,1,6,5,2,2,3,4,6,7,2,9,8,6,8,6,3,7,1,7,2,8,1,9,1,9,4,8,3,4,1,0,9,9,1,8,1,3,0,6,8,8,3,1,0,9,9,7,
A101749 ,0,0,0,3,8,8,2,0,3,2,0,3,9,2,6,7,6,6,2,4,7,2,3,2,5,2,9,8,9,8,7,0,1,4,2,7,1,1,7,8,6,2,0,4,9,4,0,0,0,5,4,2,4,6,6,0,3,3,7,8,4,3,9,0,1,9,4,8,4,8,8,7,2,3,3,3,4,3,1,2,0,7,1,4,4,9,6,8,4,6,1,9,6,3,4,0,9,0,8,3,3,0,3,7,3,
-A101750 ,2,29,293,2649,23760,215594,
+A101750 ,2,29,293,2649,23760,215594,1983334,18451711,173211045,
A101751 ,1,0,1,3,-6,32,264,-2024,2400,3420,55800,-666540,909720,2570400,90440,13101144,72406040,-3757930680,13117344800,72965762016,-261763004160,
A101752 ,1,0,1,5,-16,8,69,-767,1314,117,1774,-30405,78914,69024,
A101753 ,1,2,6,126,8598,
@@ -103213,7 +103213,7 @@ A103208 ,10,16,18,20,24,26,28,30,32,34,36,40,42,44,46,52,54,57,68,70,74,76,78,80
A103209 ,1,1,2,1,6,3,1,22,15,4,1,90,93,28,5,1,394,645,244,45,6,1,1806,4791,2380,505,66,7,1,8558,37275,24868,6345,906,91,8,1,41586,299865,272188,85405,13926,1477,120,9,1,206098,2474025,3080596,1204245,229326,26845,
A103210 ,1,3,15,93,645,4791,37275,299865,2474025,20819307,178003815,1541918901,13503125805,119352115551,1063366539315,9539785668657,86104685123025,781343125570515,7124072211203775,65233526296899981,599633539433039445,5531156299278726663,
A103211 ,1,4,28,244,2380,24868,272188,3080596,35758828,423373636,5092965724,62071299892,764811509644,9511373563492,119231457692284,1505021128450516,19112961439180588,244028820862442116,3130592301487969948,40333745806536135028,521655330655122923980,
-A103212 ,1,1,6,93,2380,85405,3956106,224939113,15175702200,1185580310121,105302043709390,10482085765658661,1156062800841590148,139945327558704629221,18449221488652046992914,
+A103212 ,1,1,6,93,2380,85405,3956106,224939113,15175702200,1185580310121,105302043709390,10482085765658661,1156062800841590148,139945327558704629221,18449221488652046992914,2631255715262150125502865,403689862107153669227378416,66297391981691913179574751633,
A103213 ,1,5,29,206,1774,18204,218868,3036144,47928816,850514400,16783812000,364865040000,8666747625600,223351748524800,6206847295622400,185007996436838400,5887506932836300800,199216094254423142400,
A103214 ,1,25,49,73,97,121,145,169,193,217,241,265,289,313,337,361,385,409,433,457,481,505,529,553,577,601,625,649,673,697,721,745,769,793,817,841,865,889,913,937,961,985,1009,1033,1057,1081,1105,1129,1153,1177,1201,
A103215 ,1,2,5,10,13,17,25,26,29,34,37,41,49,50,53,58,61,65,73,74,77,82,85,89,97,98,101,106,109,113,121,122,125,130,133,137,145,146,149,154,157,161,169,170,173,178,181,185,193,194,197,202,205,209,217,218,221,226,
@@ -106314,7 +106314,7 @@ A106309 ,5,7,11,13,17,31,37,41,53,79,107,199,
A106310 ,47,617,2693,
A106311 ,49,81,148,169,229,257,316,321,361,404,469,473,564,568,592,621,697,729,733,756,761,785,788,837,892,916,940,985,993,1016,1076,1101,1129,1229,1257,1264,1300,1304,1345,1369,1373,1384,1396,1425,1436,1489,1492,1509,
A106312 ,23,31,44,59,76,83,87,104,107,108,116,135,139,140,152,172,175,176,199,200,204,211,212,216,231,236,239,243,244,247,255,268,279,283,300,304,307,324,327,331,332,335,339,351,356,364,367,379,411,416,419,424,428,
-A106313 ,1,4,9,16,37,129,338,753,1700,3103,11587,38262,108970,314889,1052618,3214631,7956588,21949554,99877774,222744643,597394253,1932355207,7250186215,17146907277,
+A106313 ,1,4,9,16,37,129,338,753,1700,3103,11587,38262,108970,314889,1052618,3214631,7956588,21949554,99877774,222744643,597394253,1932355207,7250186215,17146907277,55160980938,155891678120,508666658005,1427745660373,
A106314 ,1,1,1,1,4,1,1,4,4,1,1,4,9,4,1,1,4,9,9,4,1,1,4,9,16,9,4,1,1,4,9,16,16,9,4,1,1,4,9,16,25,16,9,4,1,1,4,9,16,25,25,16,9,4,1,
A106315 ,0,1,2,5,4,0,6,2,1,4,10,16,12,8,12,18,16,30,18,36,20,16,22,12,13,20,28,0,28,24,30,3,36,28,44,51,36,32,44,50,40,48,42,12,36,40,46,108,33,21,60,18,52,72,4,88,68,52,58,48,60,56,66,67,8,96,66,30,84,128,70,84,72,68,78,
A106316 ,0,1,2,1,4,0,6,2,1,4,10,4,12,8,12,2,16,12,18,16,20,16,22,12,13,20,1,0,28,24,30,3,3,28,9,15,36,32,5,10,40,6,42,12,36,40,46,12,33,21,9,18,52,18,4,32,11,52,58,48,60,56,3,3,8,30,66,30,15,58,70,12,72,68,3,36,20,42,
@@ -116926,7 +116926,7 @@ A116921 ,0,1,1,1,2,1,3,3,4,3,5,5,6,5,7,7,8,7,9,9,10,9,11,11,12,11,13,13,14,13,15
A116922 ,1,1,2,3,3,5,4,5,5,7,6,7,7,9,8,9,9,11,10,11,11,13,12,13,13,15,14,15,15,17,16,17,17,19,18,19,19,21,20,21,21,23,22,23,23,25,24,25,25,27,26,27,27,29,28,29,29,31,30,31,31,33,32,33,33,35,34,35,35,37,36,37,37,39,38,
A116923 ,1,5,1,12,7,2,22,26,20,6,35,74,112,84,24,51,183,484,672,456,120,70,417,1818,4140,4968,3000,720,92,904,6288,22014,41400,42840,23040,5040,117,1900,20672,106920,295056,464040,418320,201600,40320,145,3917,65816,489696,1902960,
A116924 ,1,1,4,1,8,7,1,12,21,10,1,16,42,40,13,1,20,70,100,65,16,1,24,105,200,195,96,19,1,28,147,350,455,336,133,22,1,32,196,560,910,896,532,176,25,1,36,252,840,1638,2016,1596,792,225,28,1,40,315,1200,
-A116925 ,1,1,2,1,2,3,1,2,4,4,1,2,5,8,5,1,2,6,14,16,6,1,2,7,22,42,32,7,1,2,8,32,92,132,64,8,1,2,9,44,177,422,429,128,9,1,2,10,58,310,1122,2074,1430,256,10,1,2,11,74,506,2606,7898,10754,4862,512,11,1,2,12,92,
+A116925 ,1,1,2,1,2,3,1,2,4,4,1,2,5,8,5,1,2,6,14,16,6,1,2,7,22,42,32,7,1,2,8,32,92,132,64,8,1,2,9,44,177,422,429,128,9,1,2,10,58,310,1122,2074,1430,256,10,1,2,11,74,506,2606,7898,10754,4862,512,11,1,2,12,92,782,5462,25202,60398,58202,16796,1024,12,
A116926 ,6,14,15,51,62,91,95,159,254,287,473,679,703,1139,1199,1339,1717,1891,2051,2147,2495,2651,2701,2869,3151,4313,4381,4607,5017,5267,6245,6683,8441,9809,10063,10637,11051,11183,12403,13119,13169,13207,13423,13427,
A116927 ,0,1,0,0,1,1,0,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,2,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,2,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,1,2,0,2,0,1,0,1,0,1,2,1,1,
A116928 ,1,0,1,0,2,1,3,2,4,4,6,6,8,9,11,12,15,17,20,22,26,29,34,37,43,48,55,60,69,76,86,94,106,117,131,143,160,176,195,213,236,259,285,311,342,374,410,446,488,533,581,631,688,748,813,881,957,1038,1125,1216,1317,1425,
@@ -117395,7 +117395,7 @@ A117390 ,17,56,1888,
A117391 ,2,3,4,4,3,3,7,6,6,8,3,3,5,8,6,6,7,4,6,5,8,12,10,3,3,4,4,14,13,8,6,6,10,6,9,5,5,8,6,10,8,4,3,11,20,9,4,4,7,6,10,11,9,7,6,6,8,3,6,19,14,4,4,15,16,9,6,4,7,7,12,9,5,5,12,10,8,13,10,10,9,6,7,8,12,10,3,3,8,18,10,8,5,5,
A117392 ,7,11,47,29,23,37,157,97,199,89,127,113,317,331,839,479,293,211,541,1399,1973,1637,1129,3229,2971,3433,7253,6397,2179,3989,4297,1327,1361,4831,7963,8501,7283,4177,16301,2503,17257,7993,18839,16033,31957,9587,
A117393 ,1,2,15,65,190,447,
-A117394 ,2,2,6,30,2310,9699690,304250263527210,40729680599249024150621323470,10014646650599190067509233131649940057366334653200433090,16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999110,
+A117394 ,1,2,2,6,30,2310,9699690,304250263527210,40729680599249024150621323470,10014646650599190067509233131649940057366334653200433090,16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999110,
A117395 ,0,0,0,0,1,2,3,4,8,14,20,27,40,59,80,106,145,198,262,340,447,584,751,956,1221,1555,1959,2454,3073,3839,4760,5875,7245,8912,10909,13303,16206,19696,23848,28788,34704,41755,50085,
A117396 ,1,1,1,1,2,1,1,5,3,1,1,17,11,4,1,1,77,51,19,5,1,1,437,291,109,29,6,1,1,2957,1971,739,197,41,7,1,1,23117,15411,5779,1541,321,55,8,1,1,204557,136371,51139,13637,2841,487,71,9,1,1,2018957,1345971,504739,134597,
A117397 ,1,4,19,109,739,5779,51139,504739,5494339,65369539,843747139,11741033539,175200329539,2790549065539,47251477577539,847548190793539,16053185741897539,320165936763977539,6706533708227657539,147206624680428617539,3378708717041050697539,
@@ -120259,7 +120259,7 @@ A120254 ,0,0,0,0,1,0,1,2,1,2,2,2,3,2,5,3,4,4,5,4,5,7,5,6,6,7,9,5,9,7,8,11,8,11,8
A120255 ,1,2,3,5,4,8,13,7,21,17,34,11,55,89,6,9,12,16,18,24,36,48,72,144,233,29,377,10,61,122,305,610,47,141,329,987,1597,19,38,68,76,136,152,323,646,1292,2584,37,113,4181,15,33,41,123,165,205,451,615,1353,2255,6765,26,
A120256 ,1,0,1,1,1,2,1,2,2,2,1,10,1,2,5,4,1,10,3,11,5,2,1,55,4,2,12,11,1,52,3,8,5,2,5,133,7,4,5,46,3,52,1,27,22,6,1,260,6,40,5,11,3,100,13,78,27,6,3,874,3,4,22,48,5,52,7,27,29,116,3,1319,3,8,36,23,13,116,3,444,112,4,1,1834,
A120257 ,1,2,-1,3,-6,-1,4,-20,-20,1,5,-50,-175,70,1,6,-105,-980,1764,252,-1,7,-196,-4116,24696,19404,-924,-1,8,-336,-14112,232848,731808,-226512,-3432,1,9,-540,-41580,1646568,16818516,-24293412,-2760615,12870,1,10,-825,-108900,9343620,267227532,-1447482465,
-A120258 ,1,1,1,1,2,1,1,6,3,1,1,20,20,4,1,1,70,175,50,5,1,1,252,1764,980,105,6,1,1,924,19404,24696,4116,196,7,1,1,3432,226512,731808,232848,14112,336,8,1,1,12870,2760615,24293412,16818516,1646568,41580,540,9,1,1,48620,
+A120258 ,1,1,1,1,2,1,1,6,3,1,1,20,20,4,1,1,70,175,50,5,1,1,252,1764,980,105,6,1,1,924,19404,24696,4116,196,7,1,1,3432,226512,731808,232848,14112,336,8,1,1,12870,2760615,24293412,16818516,1646568,41580,540,9,1,
A120259 ,1,2,4,11,46,302,3109,49345,1209058,45574112,2636237374,234854695297,32081882854399,6733481882732516,2172532761103119601,1074257501384373622001,816914977299535380309346,953227711986515337529688144,1706089496424625166250326935690,
A120260 ,1,1,2,3,8,24,92,432,2740,23822,264185,3545166,59474514,1343942004,41179884383,1593533376361,74665098131246,4404743069577837,351138858279113987,37740395752334771775,5093113605218543006445,
A120261 ,1,4,10,11,13,28,17,26,31,31,20,77,28,46,67,40,28,100,26,72,120,62,32,139,44,53,71,118,32,202,35,70,135,73,97,211,33,80,130,134,36,284,45,141,183,78,50,226,68,112,150,146,38,173,150,219,182,80,38,468,36,82,
@@ -122141,7 +122141,7 @@ A122136 ,2,13,19,29,104,29,111,79,778,47,73,163,1068,359,5233,885,142,20477,219,
A122137 ,30,123,195,214,248,300,304,335,343,350,364,367,414,443,543,570,579,584,590,612,671,691,706,707,734,780,791,799,806,810,827,836,852,880,938,960,976,1015,1055,1147,1168,1190,1195,1199,1200,1210,1230,1231,1250,
A122138 ,1,2,3,4,6,8,10,11,12,14,15,18,20,22,23,26,27,32,36,38,39,40,44,47,48,50,51,52,54,55,56,58,59,60,64,66,68,71,72,74,76,78,80,83,84,86,88,89,90,92,94,95,96,98,100,102,103,107,108,110,112,114,116,118,120,122,126,
A122139 ,2,13,19,29,29,79,47,73,163,359,5233,20477,811,13859,2203,75997,3331,4457,239087,58061,159097,116041,7487,17929,4547,152657,408787,58313,5563,4783,226199,13729,676763,204641,119293,283979,2210983,7121,433,
-A122140 ,1,25,537,661,5199,113253,240621,5337048977,17434578479,34216676921,1991831965911,4495321247369,
+A122140 ,1,25,537,661,5199,113253,240621,5337048977,17434578479,34216676921,1991831965911,4495321247369,22567781434431,
A122141 ,1,1,2,1,4,0,1,6,4,0,1,8,12,0,2,1,10,24,8,4,0,1,12,40,32,6,8,0,1,14,60,80,24,24,0,0,1,16,84,160,90,48,24,0,0,1,18,112,280,252,112,96,0,4,2,1,20,144,448,574,312,240,64,12,4,0,1,22,180,672,1136,840,544,320,24,30,8,0,
A122142 ,1,25,837,5129,94375,271465,3576217,3661659,484486719,2012535795,31455148645,95748332903,145967218799,165153427677,
A122143 ,3,2,4,1,3,7,7,4,0,0,5,3,3,2,9,8,1,7,2,4,1,0,9,3,4,7,5,0,0,6,2,7,3,7,4,7,1,2,0,3,6,5,2,0,1,5,1,9,2,4,5,5,2,7,2,4,8,0,8,5,9,3,3,2,1,6,0,9,9,2,6,7,2,6,0,0,9,6,3,7,4,5,1,9,6,1,1,4,8,7,9,4,8,7,0,0,1,7,1,3,1,2,9,3,
@@ -122627,7 +122627,7 @@ A122622 ,2,3,5,7,12,16,24,28,49,67,94,118,165,175,195,268,304,367,409,498,514,56
A122623 ,2,4,8,22,98,388,16648,132358,1311234,1073743876,17179877384,274877943814,4398113693698,72083983122432004,3458834890081107976,110681871838615896070,38686511673635471463677954,2475908412805686633409019908,198070557401964757114570145800,
A122624 ,1,2,3,4,5,6,7,8,9,1024,2049,4098,8195,16388,32773,65542,131079,262152,524297,2097152,4194305,8388610,16777219,33554436,67108869,134217734,268435463,536870920,1073741833,3221225472,6442450945,12884901890,25769803779,51539607556,
A122625 ,1,2,3,4,5,6,7,8,9,100,122,146,172,200,230,262,296,332,370,800,883,970,1061,1156,1255,1358,1465,1576,1691,2700,2884,3074,3270,3472,3680,3894,4114,4340,4572,6400,6725,7058,7399,7748,8105,8470,8843,9224,9613,12500,
-A122626 ,1,4,9,22,55,114,205,388,649,10000,14884,21316,29584,40186,101705,132358,169363,315580,393130,640000,779689,940900,1400272,1662918,2345005,2746594,3721554,4308868,5661613,7290000,8317456,1073743876,1291476690,1546146538,
+A122626 ,1,4,9,22,55,114,205,388,649,10000,14884,21316,29584,40186,101705,132358,169363,315580,393130,640000,779689,940900,1400272,1662918,2345005,2746594,3721554,4308868,5661613,7290000,8317456,1073743876,1291476690,1546146538,1841269330,
A122627 ,1,2,3,4,5,6,7,8,9,55,67,80,94,109,125,142,160,179,199,420,463,508,555,604,655,708,763,820,879,1395,1489,1586,1686,1789,1895,2004,2116,2231,2349,3280,3445,3614,3787,3964,4145,4330,4519,4712,4909,6375,6631,6892,7158,
A122628 ,1,3,6,10,20,43,64,114,185,280,402,554,820,11030,14640,18910,24177,30439,37810,88410,107416,129286,154290,270000,317530,371359,431306,659350,758210,867695,988534,1394984,1576972,1775485,2383290,2665338,3459466,3846532,
A122629 ,1,2,3,4,5,6,7,8,9,55,90,146,236,381,615,993,1604,2592,4190,13530,21893,35424,57317,92740,150055,242792,392843,635630,1028467,2496120,4038808,6534929,10573737,17108665,27682400,44791062,72473458,117264515,189737967,
@@ -123376,7 +123376,7 @@ A123371 ,3,3,3,4,35,16,7,4,11,55,112,183,36,51,23,56,8,16,32,28,115,135,44,15,28
A123372 ,0,1,29,71,95,173,298,4937,4982,15755,16639,17621,
A123373 ,4,48,3598,924780,287358579128,339575512147572,836406636653653232322,2225332017808171682043720,21158384827910606570843063431876,2570789828135881020104992992114519012237948,
A123374 ,2,5,17,151,
-A123375 ,3,1,2,4,24,6,10,56,50,78,34,320,249,186,463,762,598,1238,422,760,3760,3585,9214,1765,4112,13447,6675,4585,68498,8112,10083,8650,86203,49433,35085,20641,458421,8861,366314,157857,169147,487115,277440,
+A123375 ,3,1,2,4,24,6,10,56,50,78,34,320,249,186,463,762,598,1238,422,760,3760,3585,9214,1765,4112,13447,6675,4585,68498,8112,10083,8650,86203,49433,35085,20641,458421,8861,366314,157857,169147,487115,277440,563951,511757,920602,75150,
A123376 ,28,129,1371,7141,68341,163541,624211,1086557,2756043,8546951,11791577,28122767,46308119,58262037,88870153,158512433,263952799,308206649,480993245,635060975,724715753,1053143991,1331063769,1845563079,2750645663,3325653577,3650662901,4369224195,4767074983,5637335441,
A123377 ,0,10,2916,933470,300476232,96750651250,31153377608748,10031290272012230,3230044304029586064,1040064234424568675290,334897453437128916148980,107835939942462262098571310,
A123378 ,0,60,82308,118528020,170911244616,246453665407500,355386005842019724,512466373637712510180,738976155386937410086032,1065603103601110189318267740,1536598936416627281801814920340,
@@ -128275,8 +128275,8 @@ A128270 ,2,1,-3,4,-3,12,-3,48,-3,320,-3,512,-135,256,-243,5120,-243,8192,-27,512
A128271 ,1,1,2,1,4,1,16,1,64,3,32,27,256,27,1024,243,1024,243,512,27,1024,27,1024,243,8192,243,16384,243,4096,243,7168,81,12544,243,15680,27,39200,27,62720,243,313600,243,1568000,27,31360000,27,17920000,27,31360000,27,
A128272 ,1,1,15,77,5301,189679,87596289,21608003585,68221625702463,115452529488363949,2497495662248930113941,80258100236324702562311,4295613290302749695769359713665,341566880541004135370464340131322497,
A128273 ,1,3,7,171,2401,419121,39647713,47740815747,30877916418391,255080753983140651,1130395777976404261441,177322193432863810849593,1944244855966235024678049078337,754657638581703992960984555289787011,
-A128274 ,1,1,17,721,58337,7734241,218014151,419784870961,153563504618177,10300599833780983,
-A128275 ,1,5,121,1369,698161,22394737,25947503401,29819441791,3389281372287841,354891998735343073,
+A128274 ,1,1,17,721,58337,7734241,218014151,419784870961,153563504618177,10300599833780983,2486497854930863041,30262124466958766778001,3711710439292601861342231,26350476755161831091778460321,31166629149666821954776191205937,45673109693364177065089340171611,
+A128275 ,1,5,121,1369,698161,22394737,25947503401,29819441791,3389281372287841,354891998735343073,1147649139272698443481,179707467079684030326841,76137452589606191547280211,183280927961986287722231696209,1163176750283121903011836076436361,
A128276 ,3,15,105,93081,
A128277 ,93081,449985,1523705,301921991,899343761,1581262341,7290929465,12102153569,25404516309,27482957831,38661868781,49656488021,240305617889,305000299185,341656377581,377737353491,
A128278 ,105,165,231,935,2109,2795,3021,3819,6981,7205,11285,12341,13101,16419,17549,19839,21749,21995,26391,31229,31269,46631,62651,63645,65391,76155,77585,100955,110811,113555,118031,136451,148359,150245,154679,
@@ -129978,7 +129978,7 @@ A129973 ,0,0,0,1,1,3,5,9,16,27,46,77,128,212,349,573,938,1532,2498,4067,6614,107
A129974 ,0,627,1128,2811,6188,9027,18740,38375,54908,111503,225936,322295,652152,1319115,1880736,3803283,7690628,10963995,22169420,44826527,63905108,129215111,261270408,372468527,753123120,1522797795,2170907928,
A129975 ,0,132,2295,2859,3535,15792,19060,22984,94363,113407,136275,552292,663288,796572,3221295,3868227,4645063,18777384,22547980,27075712,109444915,131421559,157811115,637894012,765983280,919792884,3717921063,
A129976 ,1,2,3,4,5,6,8,10,14,21,33,36,56,68,94,378,1943,2389,5455,
-A129977 ,2,17,102,112,316,447,535,820,1396,1475,1650,
+A129977 ,2,17,102,112,316,447,535,820,1396,1475,1650,5575,6486,6832,
A129978 ,2,3,4,5,6,7,12,16,19,21,22,25,41,114,181,236,2003,
A129979 ,1,3,3,2,3,1,3,2,2,1,
A129980 ,1,2,4,7,6,3,15,12,18,9,15,24,24,33,24,33,24,33,24,51,42,51,33,51,69,51,60,69,60,87,60,60,87,105,87,87,105,87,87,105,114,105,96,141,123,123,159,150,159,150,141,141,132,168,159,150,177,159,159,168,195,186,195,
@@ -131278,9 +131278,9 @@ A131273 ,1,23,299,313,171287,435705,487475,3774601,219347813,9613155161,51501638
A131274 ,1,295,455,4361,10817,132680789,334931875,957643538339,
A131275 ,1,17,25,31,1495,5555,8185,8647,106841,187329,345377,1811351,2179119,2863775,6368703,10250821,59137893,337430815,11349203711,183233304195,
A131276 ,1,3131,6289,323807,443371,83802527023,4076111200313,
-A131277 ,1,395191,697717,1078323,2050797,10543929,386099691,2467825171,4488040933,17387575533,39641205433,825688143387,2800262033655,3214748608393,
+A131277 ,1,395191,697717,1078323,2050797,10543929,386099691,2467825171,4488040933,17387575533,39641205433,825688143387,2800262033655,3214748608393,5174884331693,
A131278 ,1,37,265,17207,9382589,970248431,2427811793,156281194823,2955922292131,
-A131279 ,1,25,453,677,839,1015,3735,4175,4413,10369,14239,43311,452567,1274185,14102849,37801813,71271705,93524231,386557609,2151748733,261349938459,761474469415,1284262332971,
+A131279 ,1,25,453,677,839,1015,3735,4175,4413,10369,14239,43311,452567,1274185,14102849,37801813,71271705,93524231,386557609,2151748733,261349938459,761474469415,1284262332971,5115376212971,
A131280 ,4,9,14,19,22,24,27,32,37,40,45,47,50,52,57,58,62,63,65,70,75,76,83,88,90,93,95,98,100,101,103,106,108,111,113,116,124,126,129,131,133,136,138,141,142,149,151,154,159,164,167,172,174,176,177,179,182,185,190,
A131281 ,0,0,0,2,6,18,70,310,1582,9058,57678,403878,3085478,25535378,227589206,2173314806,22137209694,239580726978,2745392996254,33207657441094,422813028038230,5652593799727858,79168165551184422,1159200449070638742,17711278225214739086,
A131282 ,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,3,4,5,1,2,3,
@@ -131591,7 +131591,7 @@ A131586 ,3,30,304,3042,30424,304248,3042487,30424876,304248761,3042487612,304248
A131587 ,4,2397,8384727,19053119163,34099597499091,53251529659694763,76304519151822049179,103158861357874372432083,133759354162117403400944283,168072405102068540986037048787,206076219788796447007218742841043,247754953701579144582110673365391267,
A131588 ,1,1,3,7,11,31,43,127,171,511,683,2047,2731,8191,10923,32767,43691,131071,174763,524287,699051,2097151,2796203,8388607,11184811,33554431,44739243,134217727,178956971,536870911,715827883,2147483647,2863311531,
A131589 ,-3,3,-2,-1,9,-30,85,-229,606,-1593,4177,-10942,28653,-75021,196414,-514225,1346265,-3524574,9227461,-24157813,63245982,-165580137,433494433,-1134903166,2971215069,-7778742045,20365011070,-53316291169,139583862441,-365435296158,956722026037,
-A131590 ,4,13,87,1027,13275,155995,1789395,19523155,204330315,2081006083,20605602003,199966727443,1908356153955,17942429101363,166591116531123,1529578004981731,13917470067182067,125565110929591171,
+A131590 ,4,13,87,1027,13275,155995,1789395,19523155,204330315,2081006083,20605602003,199966727443,1908356153955,17942429101363,166591116531123,1529578004981731,13917470067182067,125565110929591171,1124685106917162579,10009134886727192611,
A131591 ,4,38,1556,86606,4083404,171658094,6716224724,247782290006,8763080657420,299863491723614,9990667099305740,325847250824377382,10445562407382412028,330039152364735149222,10301457052184951857604,318211810358946705058382,
A131592 ,4,4,87,86606,204330315,792563962432,4719861842243387,41451006295401961098,
A131593 ,4,12,34,118,520,1738,5310,13528,25332,
@@ -132119,7 +132119,7 @@ A132114 ,1,7,94,856,2995,56902,413893,1527982,47859601,702710533,1373798194,8537
A132115 ,1,2,3,4,8,9,10,12,16,22,27,28,40,42,46,52,58,64,88,100,102,106,112,130,148,162,166,172,178,184,208,268,282,292,294,328,418,424,556,562,568,586,592,598,616,640,642,646,652,658,664,688,712,784,904,1024,1072,1168,1240,
A132116 ,1,1,4,2,1,2,3,7,3,3,30,2,1,2,2,83,9,20,1,37,1,2,7,1,1,2,1,6,1,2,1,1,3,3,1,4,8,1,6,33,1,1,1,17,4,1,3,1,5,3,2,1,1100,2,31,6,7,1,1,9,6,3,1,2,2,2,1,2,4,6,16,1,1,8,1,13,2,18,1,4,1,46,2,5,1,3,1,42,1,1,1,26,3,2,1,5,4,
A132117 ,1,8,32,90,205,406,728,1212,1905,2860,4136,5798,7917,10570,13840,17816,22593,28272,34960,42770,51821,62238,74152,87700,103025,120276,139608,161182,185165,211730,241056,273328,308737,347480,389760,435786,485773,539942,598520,
-A132118 ,1,2,4,4,6,8,7,9,11,13,11,13,15,17,19,16,18,20,22,24,26,22,24,26,28,30,32,34,29,31,33,35,37,39,41,43,37,39,41,43,45,47,49,51,53,46,48,50,52,54,56,58,60,62,64,
+A132118 ,1,2,4,4,6,8,7,9,11,13,11,13,15,17,19,16,18,20,22,24,26,22,24,26,28,30,32,34,29,31,33,35,37,39,41,43,37,39,41,43,45,47,49,51,53,46,48,50,52,54,56,58,60,62,64,56,58,60,62,64,66,68,70,72,74,76,
A132119 ,1,3,3,6,5,6,10,8,9,10,15,12,13,14,15,21,17,18,19,20,21,28,23,24,25,26,27,28,36,30,31,32,33,34,35,36,45,38,39,40,41,42,43,44,45,55,47,48,49,50,51,52,53,54,55,
A132120 ,3,6,2,3,0,6,2,2,2,3,6,6,4,9,8,0,4,8,7,9,8,6,2,6,3,7,2,2,2,4,0,9,3,4,6,1,8,1,1,1,7,9,8,5,8,5,3,4,4,2,0,9,9,9,7,5,9,9,5,1,0,1,7,0,2,7,8,4,1,8,8,6,3,0,6,8,9,6,5,0,
A132121 ,0,1,2,5,11,17,14,32,50,68,30,70,110,150,190,55,130,205,280,355,430,91,217,343,469,595,721,847,140,336,532,728,924,1120,1316,1512,204,492,780,1068,1356,1644,1932,2220,2508,285,690,1095,1500,1905,2310,2715,3120,
@@ -132128,7 +132128,7 @@ A132123 ,0,11,110,469,1356,3135,6266,11305,18904,29811,44870,65021,91300,124839,
A132124 ,0,3,17,50,110,205,343,532,780,1095,1485,1958,2522,3185,3955,4840,5848,6987,8265,9690,11270,13013,14927,17020,19300,21775,24453,27342,30450,33785,37355,41168,45232,49555,54145,59010,64158,69597,75335,81380,
A132125 ,1,2,3,4,5,6,7,7,7,7,8,8,9,9,9,9,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,
A132126 ,0,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
-A132127 ,1,6,17,37,69,116,181,267,377,514,681,881,
+A132127 ,1,6,17,37,69,116,181,267,377,514,681,881,1117,1392,1709,2071,2481,2942,3457,4029,4661,5356,6117,6947,7849,8826,9881,11017,12237,13544,14941,16431,18017,19702,21489,23381,25381,27492,29717,32059,34521,37106,39817,42657,45629,
A132128 ,1,2,4,4,5,8,7,8,9,13,11,12,13,14,19,16,17,18,19,20,26,22,23,24,25,26,27,34,29,30,31,32,33,34,35,43,37,38,39,40,41,42,43,44,53,
A132129 ,2,19,19,577,7417,114229,2053313,42373937,987654103,25678048763,736867805209,23136292864193,789018236128391,29043982525257901,1147797409030815779,48471109094902530293,2178347851919531491093,103805969587115219167613,5228356786703601108008083,
A132130 ,1,6,21,62,162,378,819,1680,3276,6138,11145,19662,33840,57048,94362,153432,245757,388218,605466,933414,1423614,2149586,3215844,4769544,7016572,10243896,14848809,21378276,30582360,43484304,61473438,86428896,
@@ -133757,7 +133757,7 @@ A133752 ,1,256,65536,16777216,4294967296,1099511627776,281474976710656,720575940
A133753 ,73,733,3733,7253,7523,7757,33223,35323,72253,72353,73327,73523,73553,75223,75253,77237,77323,77527,77557,333323,352333,355723,375223,375233,375553,722333,727327,733333,735733,737353,737753,737773,753373,753527,772273,773273,775757,777353,777373,
A133754 ,0,0,24,216,960,3000,7560,16464,32256,58320,99000,159720,247104,369096,535080,756000,1044480,1414944,1883736,2469240,3192000,4074840,5142984,6424176,7948800,9750000,11863800,14329224,17188416,20486760,24273000,28599360,33521664,
A133755 ,0,0,0,1,1,1,2,1,2,2,3,1,4,2,3,3,5,2,6,2,4,4,7,3,7,4,6,4,9,2,10,5,7,6,8,4,12,6,8,6,13,4,14,6,8,8,15,5,14,6,11,8,17,6,14,8,12,10,19,6,20,10,12,11,16,6,22,10,15,8,23,8,24,12,13,12,20,8,26,10,18,14,27,8,22,14,19,14,
-A133756 ,5,7,10,14,21,33,52,85,144,248,438,791,1456,2731,5213,10112,19920,39819,80704,165749,344758,725888,1546398,3331879,
+A133756 ,5,7,10,14,21,33,52,85,144,248,438,791,1456,2731,5213,10112,19920,39819,80704,165749,344758,725888,1546398,3331879,7257856,15978098,35538283,79834112,181082105,414609369,958004224,2233339296,5251710002,12454043648,29777842663,71773351064,174356586496,426815713006,1052675774422,
A133757 ,0,1,2,4,9,7,20,23,27,
A133758 ,0,3,5,4,149,7,144,37,1442,
A133759 ,2,5,6,9,10,13,14,16,17,21,23,26,28,30,31,36,37,38,39,41,44,47,48,50,51,52,54,55,58,60,61,65,67,69,71,74,76,79,82,84,86,87,93,95,96,99,100,101,103,105,106,108,112,115,116,117,118,119,121,122,126,128,132,133,134,
@@ -137230,7 +137230,7 @@ A137225 ,1,3,2,7,5,3,15,8,7,4,31,17,11,9,5,63,26,15,14,11,6,127,53,31,19,17,13,7
A137226 ,29,41,71,101,149,197,239,269,311,419,461,521,599,617,641,809,821,827,881,1031,1061,1151,1289,1427,1607,1697,1721,1871,2267,2381,2657,2687,2789,2969,3251,3299,3527,3539,3581,3821,3929,4001,4049,4091,4229,4241,
A137227 ,1,1,1,2,2,2,6,9,9,22,24,64,64,266,708,120,625,625,4536,17457,108129,720,7776,7776,100392,563088,5709120,52517688,5040,117649,117649,2739472,22516209,375217945,5489293264,92757410569,40320,2097152,
A137228 ,3,5,7,9,11,12,14,16,18,19,21,23,24,26,28,29,31,33,34,36,38,39,41,42,44,46,47,49,51,52,54,55,57,59,60,62,63,65,67,68,70,71,73,75,76,78,79,81,83,84,86,87,89,90,92,94,95,97,98,100,102,103,105,106,108,109,111,113,
-A137229 ,1,4,11,27,64,150,350,815,1896,4409,10251,23832,55404,128800,299425,696080,1618191,3761839,8745216,20330162,47261894,109870575,255418100,593775045,1380359511,3208946544,7459895656,17342153392,40315615409,
+A137229 ,1,4,11,27,64,150,350,815,1896,4409,10251,23832,55404,128800,299425,696080,1618191,3761839,8745216,20330162,47261894,109870575,255418100,593775045,1380359511,3208946544,7459895656,17342153392,40315615409,93722435100,217878227875,
A137230 ,4,6,10,12,14,16,18,22,24,26,27,30,34,36,38,40,42,45,46,56,58,60,62,63,66,74,75,78,80,82,84,86,88,94,96,99,100,102,104,105,106,114,117,118,120,122,132,134,136,138,140,142,144,146,147,152,153,156,158,165,166,
A137231 ,6552,30240,70680,87360,120960,120960,120960,138240,157248,157248,161280,196560,211680,229320,241920,241920,241920,241920,262080,280800,290160,302400,338688,362880,362880,393120,393120,446400,446400,483840,
A137232 ,0,0,1,-1,8,-12,65,-125,544,-1224,4657,-11593,40520,-107700,356561,-988901,3161728,-9014352,28179745,-81795025,252010184,-740036124,2258722337,-6682944653,20273892640,-60278338200,182146752721,-543273442201,1637465696648,-4893939533892,14726379083825,
@@ -137260,8 +137260,8 @@ A137255 ,1,2,4,8,17,36,80,178,409,942,2212,5204,12377,29472,70592,169198,406801,
A137256 ,1,2,4,9,21,48,108,243,549,1242,2808,6345,14337,32400,73224,165483,373977,845154,1909980,4316409,9754749,22044960,49819860,112588947,254442141,575019162,1299497904,2936762649,6636851721,14998760928,
A137257 ,4,12,16,18,20,24,27,28,36,44,48,50,52,54,60,64,68,72,76,80,84,90,92,98,100,108,112,116,120,124,126,132,135,140,144,148,150,156,160,162,164,168,172,176,180,188,189,192,196,198,200,204,208,212,216,220,228,234,
A137258 ,3,5,7,17,19,139,157,577,1201,27361,530401,2513281,7183201,407817217,
-A137259 ,0,0,0,3,3,0,20,20,16,0,115,115,110,90,0,714,714,708,684,576,0,5033,5033,5026,4998,4872,4200,0,40312,40312,40304,40272,40128,39360,34560,0,362871,362871,362862,362826,362664,361800,356400,317520,0,3628790,
-A137260 ,0,0,0,1,2,0,5,10,12,0,23,46,66,72,0,119,238,354,456,480,0,719,1438,2154,2856,3480,3600,0,5039,10078,15114,20136,25080,29520,30240,0,40319,80638,120954,161256,201480,241200,277200,282240,0,362879,725758,
+A137259 ,0,0,0,3,3,0,20,20,16,0,115,115,110,90,0,714,714,708,684,576,0,5033,5033,5026,4998,4872,4200,0,40312,40312,40304,40272,40128,39360,34560,0,362871,362871,362862,362826,362664,361800,356400,317520,0,3628790,3628790,3628780,3628740,3628560,3627600,3621600,3578400,3225600,0,
+A137260 ,0,0,0,1,2,0,5,10,12,0,23,46,66,72,0,119,238,354,456,480,0,719,1438,2154,2856,3480,3600,0,5039,10078,15114,20136,25080,29520,30240,0,40319,80638,120954,161256,201480,241200,277200,282240,0,362879,725758,1088634,1451496,1814280,2176560,2535120,2862720,2903040,0,
A137261 ,1200,360,360,29172,360,360,360,5765161,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,360,
A137262 ,1,22,671,21097,666716,21082008,666667166,21081852648,666666671666,21081851083600,
A137263 ,26,50,56,86,134,170,176,236,254,260,266,356,386,446,473,506,515,560,566,590,596,650,656,680,803,944,950,974,980,1016,1100,1106,1184,1190,1220,1226,1268,1286,1313,1364,1370,1436,1496,1505,1517,1556,1604,1616,
@@ -138170,7 +138170,7 @@ A138165 ,104869,108649,140689,140869,148609,164089,164809,168409,184609,186049,4
A138166 ,1,12,20,21,22,23,24,25,26,27,28,29,32,42,52,62,72,82,92,103,113,123,130,131,132,133,134,135,136,137,138,139,143,153,163,173,183,193,203,213,223,230,231,232,233,234,235,236,237,238,239,243,253,263,273,283,
A138167 ,1,5,6,7,8,9,10,11,12,21,31,36,37,38,39,40,41,42,43,44,49,58,66,67,68,76,86,95,96,97,98,104,113,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,
A138168 ,1,2,6,7,8,9,12,20,21,22,23,26,27,29,38,44,45,46,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,71,78,79,87,92,93,94,95,103,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,
-A138169 ,1,0,-2,2,-1,1,6,-12,6,0,12,-24,-12,72,-72,24,24,-52,-88,356,-240,-360,720,-480,120,0,-720,2280,-1320,-3720,6360,-1200,-6000,7200,-3600,720,-3060,10260,2580,-56340,86760,-12480,-95760,93240,12600,-88200,75600,-30240,5040,0,171360,-745920,994560,383040,
+A138169 ,1,0,-2,2,-1,1,6,-12,6,0,12,-24,-12,72,-72,24,24,-52,-88,356,-240,-360,720,-480,120,0,-720,2280,-1320,-3720,6360,-1200,-6000,7200,-3600,720,-3060,10260,2580,-56340,86760,-12480,-95760,93240,12600,-88200,75600,-30240,5040,
A138170 ,2,3,5,23,31,61,83,89,149,179,239,251,263,269,353,367,419,433,449,503,557,569,571,587,653,701,733,761,839,941,983,991,1109,1123,1187,1193,
A138171 ,45,81,105,117,165,225,261,273,297,315,325,333,345,357,385,405,435,441,465,477,495,513,525,555,561,567,585,595,621,625,627,651,675,693,705,715,765,777,795,801,825,837,855,861,885,891,897,915,925,945,957,975,
A138172 ,62,74,134,146,164,188,194,206,254,274,278,284,314,356,362,386,398,404,422,428,454,458,482,494,524,538,554,566,584,614,626,662,674,692,734,746,758,764,794,818,824,854,866,890,914,926,934,944,956,974,998,1004,1028,
@@ -138743,9 +138743,9 @@ A138738 ,1,1,1,7,25,121,1561,9871,101137,1293265,15765841,226501111,3355388521,5
A138739 ,1,1,2,11,88,888,10572,143214,2159154,35702442,640873656,12394383780,256762580460,5671209169168,133041670286160,3304034094162183,86616702087692256,2390831825522972392,69323685702986714272,2107073248164657741448,67003070810599639419680,2225053954972969636237280,77034579373254666948386880,2776183496539544726567249520,
A138740 ,1,1,-2,9,-56,420,-3572,33328,-334354,3559310,-39838760,465743720,-5658983108,71191948512,-924554859776,12365546196641,-169995491295312,2398380272232272,-34680290150700800,513390937937217088,-7773229533145403728,
A138741 ,1,3,2,0,1,0,2,6,2,0,0,0,3,3,2,0,0,0,2,6,2,0,2,0,1,6,2,0,0,0,2,0,4,0,0,0,2,9,0,0,1,0,4,6,2,0,0,0,2,0,2,0,0,0,2,6,2,0,2,0,1,6,4,0,0,0,0,6,2,0,0,0,4,3,2,0,2,0,2,6,0,0,0,0,3,0,2,
-A138742 ,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,2,4,23,
-A138743 ,1,1,1,3,6,6,
-A138744 ,1,1,2,4,8,33,
+A138742 ,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,2,4,23,1,1,2,2,9,1,90,1,14,5,2,1,1,2,2,7,1,2,4,5,1,2,4,1,8,32,2,1,8,3,1,2,1,8,5,2,3,1,1,2,2,8,11,4,3,3,2,3,4,3,8,1,6,22,4,2,1,1,1,1,1,5,1,1,2,2,1,11,1,4,3,3,97,3,1,1,4,1,1,3,87,5,2,7,3,
+A138743 ,1,1,1,3,6,6,11,26,48,82,201,379,836,1554,3197,6420,12639,25298,50675,101675,203379,405946,811519,1622692,3249540,6494117,12998399,25991681,
+A138744 ,1,1,2,4,8,33,128,109,344,3760,1829,18367,11168,35246,41103,79356,151643,344725,1249071,1678788,5385320,19780986,17348076,30966961,85647848,160394455,451333739,623813606,
A138745 ,1,-1,1,-3,1,-2,3,0,1,-1,2,0,3,-2,0,-6,1,-2,1,0,2,0,0,0,3,-3,2,-3,0,-2,6,0,1,0,2,0,1,-2,0,-6,2,-2,0,0,0,-2,0,0,3,-1,3,-6,2,-2,3,0,0,0,2,0,6,-2,0,0,1,-4,0,0,2,0,0,0,1,-2,2,-9,0,0,6,0,
A138746 ,1,-1,3,-1,2,-3,0,-1,1,-2,0,-3,2,0,6,-1,2,-1,0,-2,0,0,0,-3,3,-2,3,0,2,-6,0,-1,0,-2,0,-1,2,0,6,-2,2,0,0,0,2,0,0,-3,1,-3,6,-2,2,-3,0,0,0,-2,0,-6,2,0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,9,0,0,-6,
A138747 ,1,1,1,2,6,19,61,197,637,2060,6662,21545,69677,225337,728745,2356778,7621874,24649315,79716449,257804821,833746693,2696355892,8720076682,28200927617,91202445513,294950796673,953877628705,3084862088210,9976514614558,32264276654339,104343409321397,337448974463477,1091317708583837,3529346452933372,11413987225587534,
@@ -140341,7 +140341,7 @@ A140336 ,1,5,16,47,138,419,1358,4615,16562,61879,240506,967043,4011246,17127819,
A140337 ,1,8,161,3771,65536,968434,13398995,190804729,2840301338,
A140338 ,1,8,256,23045,3499498,540106922,68719476736,
A140339 ,1,8,256,32768,13213313,11522939359,14149679017875,
-A140340 ,29,59,109,179,269,379,509,659,829,1019,1229,1459,1709,1979,76,86,116,166,236,326,436,566,716,886,1076,1286,1516,1766,161,151,161,191,241,311,401,511,641,791,961,1151,1361,1591,284,254,244,254,284,
+A140340 ,29,59,109,179,269,379,509,659,829,1019,1229,1459,1709,1979,76,86,116,166,236,326,436,566,716,886,1076,1286,1516,1766,161,151,161,191,241,311,401,511,641,791,961,1151,1361,1591,284,254,244,254,284,334,404,494,604,734,884,1054,1244,1454,
A140341 ,1,4,4,5,5,5,5,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,
A140342 ,0,0,0,0,0,1,5,14,28,42,42,0,-131,-417,-924,-1652,-2380,-2380,0,7753,25213,56714,102256,147798,147798,0,-479779,-1557649,-3499720,-6305992,-9112264,-9112264,0,29587889,96072133,215873462,388991876,562110290,562110290,0,
A140343 ,0,0,0,0,1,4,9,14,14,0,-41,-122,-243,-364,-364,0,1093,3280,6561,9842,9842,0,-29525,-88574,-177147,-265720,-265720,0,797161,2391484,4782969,7174454,7174454,0,-21523361,-64570082,-129140163,-193710244,-193710244,0,581130733,
@@ -141389,7 +141389,7 @@ A141384 ,8,8,32,158,828,4408,23564,126106,675076,3614144,19349432,103593806,5546
A141385 ,3,7,31,157,827,4407,23563,126105,675075,3614143,19349431,103593805,554625899,2969386479,15897666067,85113810057,455687062275,2439682811479,13061709929935,69930511268509,374397872321627,
A141386 ,3,11,12,27,129,138,273,
A141387 ,0,1,1,2,4,2,3,7,7,3,4,10,12,10,4,5,13,17,17,13,5,6,16,22,24,22,16,6,7,19,27,31,31,27,19,7,8,22,32,38,40,38,32,22,8,9,25,37,45,49,49,45,37,25,9,10,28,42,52,58,60,58,52,42,28,10,
-A141388 ,0,6,6,23,60,23,56,208,208,56,109,496,713,496,109,184,968,1696,1696,968,184,279,1664,3311,4032,3311,1664,279,384,2616,5704,7872,7872,5704,2616,384,473,3840,9005,13568,15369,13568,9005,3840,473,488,5320,13312,21440,
+A141388 ,0,6,6,23,60,23,56,208,208,56,109,496,713,496,109,184,968,1696,1696,968,184,279,1664,3311,4032,3311,1664,279,384,2616,5704,7872,7872,5704,2616,384,473,3840,9005,13568,15369,13568,9005,3840,473,488,5320,13312,21440,26488,26488,21440,13312,5320,488,
A141389 ,2,4,9,7,12,1,5,6,11,13,15,8,3,18,21,10,16,25,24,27,20,29,14,30,26,19,28,35,32,33,22,38,17,37,36,42,23,34,41,47,43,39,46,31,40,45,54,55,51,57,58,44,52,56,49,50,60,53,65,66,59,68,63,48,71,62,70,74,69,73,77,64,76,78,61,79,81,82,85,86,
A141390 ,781,1541,5461,13021,15751,25351,29539,38081,40501,79381,100651,121463,133141,195313,216457,315121,318551,319507,326929,341531,353827,375601,416641,432821,453331,464881,498451,555397,556421,753667,764941,863329,872101,886411,
A141391 ,1,7,5,11,28,14,182,70,2,66,1518,462,1540,616,296,600,1950,750,10,730,2336,876,2436,996,3154,1162,5698,210,1554,3234,1638,5382,1872,23088,4368,5934,201,4359,77991,7021,13090,4270,12950,74,12802,76466,16954,
@@ -142466,17 +142466,17 @@ A142461 ,1,1,1,1,14,1,1,111,111,1,1,796,2886,796,1,1,5597,52642,52642,5597,1,1,3
A142462 ,1,1,1,1,16,1,1,143,143,1,1,1166,4290,1166,1,1,9357,90002,90002,9357,1,1,74892,1621383,3960088,1621383,74892,1,1,599179,27016857,134142043,134142043,27016857,599179,1,1,4793482,431017552,3923731798,7780238494,
A142463 ,-1,3,11,23,39,59,83,111,143,179,219,263,311,363,419,479,543,611,683,759,839,923,1011,1103,1199,1299,1403,1511,1623,1739,1859,1983,2111,2243,2379,2519,2663,2811,2963,3119,3279,3443,3611,3783,3959,4139,4323,4511,4703,4899,5099,
A142464 ,3,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
-A142465 ,1,1,1,1,7,1,1,28,28,1,1,84,336,84,1,1,210,2520,2520,210,1,1,462,13860,41580,13860,462,1,1,924,60984,457380,457380,60984,924,1,1,1716,226512,3737448,9343620,3737448,226512,1716,1,1,3003,736164,24293412,133613766,
+A142465 ,1,1,1,1,7,1,1,28,28,1,1,84,336,84,1,1,210,2520,2520,210,1,1,462,13860,41580,13860,462,1,1,924,60984,457380,457380,60984,924,1,1,1716,226512,3737448,9343620,3737448,226512,1716,1,1,3003,736164,24293412,133613766,133613766,24293412,736164,3003,1,
A142466 ,127,227,277,577,677,727,827,877,977,1277,1327,1427,1627,1777,1877,2027,2377,2477,2677,2777,2927,3527,3677,3727,3877,4027,4127,4177,4327,4877,5077,5227,5477,5527,5827,5927,6277,6427,6577,6827,6977,7027,7127,7177,
-A142467 ,1,1,1,1,8,1,1,36,36,1,1,120,540,120,1,1,330,4950,4950,330,1,1,792,32670,108900,32670,792,1,1,1716,169884,1557270,1557270,169884,1716,1,1,3432,736164,16195608,44537922,16195608,736164,3432,1,1,6435,2760615,
+A142467 ,1,1,1,1,8,1,1,36,36,1,1,120,540,120,1,1,330,4950,4950,330,1,1,792,32670,108900,32670,792,1,1,1716,169884,1557270,1557270,169884,1716,1,1,3432,736164,16195608,44537922,16195608,736164,3432,1,
A142468 ,1,1,1,1,9,1,1,45,45,1,1,165,825,165,1,1,495,9075,9075,495,1,1,1287,70785,259545,70785,1287,1,1,3003,429429,4723719,4723719,429429,3003,1,1,6435,2147145,61408347,184225041,61408347,2147145,6435,1,
A142469 ,3,3,46,6,46,347,532,532,347,1932,14505,740,14505,1932,9199,203925,152405,152405,203925,9199,40250,2087884,6882086,-86372,6882086,2087884,40250,168318,17968725,152844537,78623775,78623775,152844537,17968725,168318,
-A142470 ,1,1,1,1,8,1,1,30,30,1,1,80,300,80,1,1,175,1750,1750,175,1,1,336,7350,19600,7350,336,1,1,588,24696,144060,144060,24696,588,1,1,960,70560,790272,1728720,790272,70560,960,1,1,1485,178200,3492720,14669424,
+A142470 ,1,1,1,1,8,1,1,30,30,1,1,80,300,80,1,1,175,1750,1750,175,1,1,336,7350,19600,7350,336,1,1,588,24696,144060,144060,24696,588,1,1,960,70560,790272,1728720,790272,70560,960,1,1,1485,178200,3492720,14669424,14669424,3492720,178200,1485,1,
A142471 ,0,0,2,2,6,14,86,1206,103718,125083910,12973452977382,1622770224612082123622,21052933202100473722674133293917606,34164073141115747076263787631563122725393126176374288934,
-A142472 ,1,-4,1,21,-18,1,-140,240,-48,1,1140,-3150,1300,-100,1,-11004,43620,-29700,4800,-180,1,123074,-650769,647780,-175175,13965,-294,1,-1566928,10517108,-14190400,5676160,-764400,34496,-448,1,22390488,-184052520,319680732,-175091112,35160048,-2698920,75600,
-A142473 ,1,-1,2,4,-6,6,-36,44,-36,24,576,-600,420,-240,120,-14400,13152,-8100,4080,-1800,720,518400,-423360,233856,-105840,42000,-15120,5040,-25401600,18817920,-9455040,3898944,-1411200,463680,-141120,40320,1625702400,-1104606720,510295680,-193777920,64653120,
+A142472 ,1,-4,1,21,-18,1,-140,240,-48,1,1140,-3150,1300,-100,1,-11004,43620,-29700,4800,-180,1,123074,-650769,647780,-175175,13965,-294,1,-1566928,10517108,-14190400,5676160,-764400,34496,-448,1,22390488,-184052520,319680732,-175091112,35160048,-2698920,75600,-648,1,
+A142473 ,1,-1,2,4,-6,6,-36,44,-36,24,576,-600,420,-240,120,-14400,13152,-8100,4080,-1800,720,518400,-423360,233856,-105840,42000,-15120,5040,-25401600,18817920,-9455040,3898944,-1411200,463680,-141120,40320,1625702400,-1104606720,510295680,-193777920,64653120,-19595520,5503680,-1451520,362880,
A142474 ,1,0,1,2,4,9,19,41,88,189,406,872,1873,4023,8641,18560,39865,85626,183916,395033,848491,1822473,3914488,8407925,18059374,38789712,83316385,178955183,384377665,825604416,1773314929,3808901426,8181135700,17572253481,37743426307,
-A142475 ,1,0,0,-1,0,0,1,-1,0,0,0,1,-1,0,0,-1,-1,1,-1,0,0,1,2,-1,1,-1,0,0,0,-3,1,-1,1,-1,0,0,-1,4,0,1,-1,1,-1,0,0,1,-6,-1,-1,1,-1,1,-1,0,0,0,9,2,2,-1,1,-1,1,-1,0,0,-1,-13,-3,-3,1,-1,1,-1,1,-1,0,0,1,19,3,4,0,1,-1,1,-1,1,-1,0,0,0,-28,-2,-5,-1,-1,1,-1,1,-1,1,-1,0,0,-1,
+A142475 ,1,0,0,-1,0,0,1,-1,0,0,0,1,-1,0,0,-1,-1,1,-1,0,0,1,2,-1,1,-1,0,0,0,-3,1,-1,1,-1,0,0,-1,4,0,1,-1,1,-1,0,0,1,-6,-1,-1,1,-1,1,-1,0,0,0,9,2,2,-1,1,-1,1,-1,0,0,-1,-13,-3,-3,1,-1,1,-1,1,-1,0,0,1,19,3,4,0,1,-1,1,-1,1,-1,0,0,0,-28,-2,-5,-1,-1,1,-1,1,-1,1,-1,0,0,
A142476 ,103,307,409,613,919,1021,1123,1327,1429,1531,2143,2347,2551,2857,3061,3163,3469,3571,3673,3877,4591,4999,5101,5407,6121,6427,6529,6733,7039,7243,7549,7753,8059,8161,8263,8467,9181,9283,10099,10303,10711,11119,11527,
A142477 ,2,53,257,359,461,563,971,1277,1481,1583,1787,1889,2297,2399,2909,3011,3623,3929,4133,4337,4643,5051,5153,5867,6173,6581,7193,7499,7703,7907,8009,8111,9029,9437,9539,9743,10151,10253,10457,10559,11069,11171,11273,
A142478 ,157,463,769,1279,1381,1483,1789,1993,2503,2707,3217,3319,3727,3931,4339,4441,4951,5563,5869,6073,6277,6379,6481,6991,7297,7603,8011,8317,8419,8521,8623,8929,9133,9337,9439,9643,9949,10357,10459,10663,10867,11071,
@@ -142597,7 +142597,7 @@ A142592 ,29,83,137,191,353,461,569,677,839,947,1109,1163,1217,1433,1487,1811,197
A142593 ,1,2,1,3,1,4,1,1,5,1,1,6,1,1,1,7,1,1,1,8,3,1,1,9,3,1,1,10,3,1,1,1,11,3,1,1,1,12,3,1,1,1,1,13,3,1,1,1,1,14,3,1,1,1,1,15,3,1,1,1,1,16,3,1,1,1,1,1,17,3,1,1,1,1,1,18,3,1,1,1,1,1,1,19,3,1,1,1,1,1,1,20,3,1,1,
A142594 ,1,2,3,4,5,6,7,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,288,300,390,405,420,435,450,465,480,495,510,525,540,555,570,585,600,615,630,645,660,675,690,705,1728,1764,1800,
A142595 ,1,1,1,1,4,1,1,10,10,1,1,22,40,22,1,1,46,124,124,46,1,1,94,340,496,340,94,1,1,190,868,1672,1672,868,190,1,1,382,2116,5080,6688,5080,2116,382,1,1,766,4996,14392,23536,23536,14392,4996,766,1,
-A142596 ,1,1,1,1,6,1,1,21,21,1,1,66,126,66,1,1,201,576,576,201,1,1,606,2331,3456,2331,606,1,1,1821,8811,17361,17361,8811,1821,1,1,5466,31896,78516,104166,78516,31896,5466,1,1,16401,112086,331236,548046,548046,331236,
+A142596 ,1,1,1,1,6,1,1,21,21,1,1,66,126,66,1,1,201,576,576,201,1,1,606,2331,3456,2331,606,1,1,1821,8811,17361,17361,8811,1821,1,1,5466,31896,78516,104166,78516,31896,5466,1,1,16401,112086,331236,548046,548046,331236,112086,16401,1,
A142597 ,1,1,1,1,8,1,1,36,36,1,1,148,288,148,1,1,596,1744,1744,596,1,1,2388,9360,13952,9360,2388,1,1,9556,46992,93248,93248,46992,9556,1,1,38228,226192,560960,745984,560960,226192,38228,1,1,152916,1057680,3148608,
A142598 ,1,1,0,1,0,1,1,0,1,1,1,0,1,1,0,1,0,1,1,0,1,1,0,1,1,0,2,1,1,0,1,1,0,2,1,0,1,0,1,1,0,2,1,1,1,1,0,1,1,0,2,1,2,2,1,1,0,1,1,0,2,1,2,2,1,0,1,0,1,1,0,2,1,2,2,2,2,1,1,0,1,1,0,2,1,2,2,2,3,2,1,1,0,1,1,0,2,1,2,2,2,3,2,2,0,
A142599 ,5,2,77,35,221,20,437,143,725,56,1085,323,1517,110,2021,575,2597,182,3245,899,3965,272,4757,1295,5621,380,6557,1763,7565,506,8645,2303,9797,650,11021,2915,12317,812,13685,3599,15125,992,16637,4355,18221,1190,19877,5183,21605,1406,23405,
@@ -155876,7 +155876,7 @@ A155871 ,1,1,-16,-110,-16,-117,-1322,-1322,-117,-512,-9703,-22288,-9703,-512,-18
A155872 ,2,21,221,2331,24641,261051,2771561,29487171,314358881,3357947691,35937424601,385311670611,4138428376721,44522712143931,479749833583241,5177248169415651,55949729863572161,605447028499293771,
A155873 ,2,3,5,7,23,37,59,73,233,373,593,739,2339,3733,5939,7393,23399,37337,59393,73939,233993,373379,593933,739391,2339933,3733799,5939333,7393913,23399339,37337999,59393339,73939133,
A155874 ,4,8,4,6,4,4,4,8,4,6,4,4,4,8,4,6,4,4,4,6,4,4,4,4,4,8,4,6,4,4,4,4,4,6,4,4,4,8,4,6,4,4,4,6,4,4,4,4,4,6,4,4,4,4,4,8,4,6,4,4,4,4,4,6,4,4,4,8,4,6,4,4,4,4,4,6,4,4,4,6,4,4,4,4,4,6,4,4,4,4,4,4,4,6,4,4,4,8,4,6,4,
-A155875 ,4,9,6,9,8,9,10,15,12,15,14,15,16,21,18,21,20,21,22,25,24,
+A155875 ,4,9,6,9,8,9,10,15,12,15,14,15,16,21,18,21,20,21,22,25,24,25,26,27,28,33,30,33,32,33,34,35,36,39,38,39,40,45,42,45,44,45,46,49,48,49,50,51,52,55,54,55,56,57,58,63,60,63,62,63,64,65,66,69,68,69,70,
A155876 ,1,1,3,0,8,9,0,4,8,9,0,1,4,2,5,3,2,1,8,0,7,3,5,3,7,0,6,1,0,4,0,6,6,7,2,1,1,1,2,4,7,0,6,2,8,1,3,6,0,6,3,8,7,6,4,8,7,3,6,1,0,9,3,3,0,0,3,6,9,4,5,3,6,3,9,0,6,9,1,1,7,4,0,2,7,6,8,8,6,3,1,4,6,5,5,2,2,0,5,
A155877 ,9,11,13,15,23,25,27,37,39,51,263,265,267,277,279,291,517,519,531,771,65543,65545,65547,65557,65559,65571,65797,65799,65811,66051,131077,131079,131091,131331,196611,4294967303,4294967305,4294967307,
A155878 ,4,8,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,39,40,42,44,45,46,48,49,51,52,54,55,57,58,60,62,63,64,65,66,69,70,72,74,75,76,77,78,80,82,84,86,87,88,90,91,93,94,95,96,98,99,102,104,105,106,108,
@@ -156655,7 +156655,7 @@ A156650 ,85,89,91,101,119,145,175,185,221,289,349,371,461,595,769,959,1021,1241,
A156651 ,1,-2,-65,227,25285,-151321,-24851645,208985867,45675130345,-494053998001,-134944451180825,1784108230326707,584753480627757805,-9136756258815374281,-3493729771694615374805,62987838852907115030747,27526170759461264661539665,
A156652 ,1,-1,-77,239,30877,-160801,-30468977,222359759,56025500377,-525750911041,-165532771357877,1898604115708079,717305800978797877,-9723130520022672481,-4285693661775748922777,67030256200148854573199,33765846794176822397603377,
A156653 ,1,1,3,1,16,13,1,125,171,39,1,1296,2551,1091,101,1,16807,43653,28838,5498,243,1,262144,850809,780585,243790,24270,561,1,4782969,18689527,22278189,10073955,1733035,98661,1263,1,
-A156654 ,1,3,1,25,22,1,343,515,101,1,6561,14156,5766,396,1,161051,456197,299342,49642,1447,1,4826809,16985858,15796159,4592764,371239,5090,1,170859375,719818759,878976219,383355555,58474285,2550165,17481,1,6975757441,
+A156654 ,1,3,1,25,22,1,343,515,101,1,6561,14156,5766,396,1,161051,456197,299342,49642,1447,1,4826809,16985858,15796159,4592764,371239,5090,1,170859375,719818759,878976219,383355555,58474285,2550165,17481,1,6975757441,34264190872,52246537948,31191262504,7488334150,660394024,16574428,59032,1,
A156655 ,3001,4001,7001,9001,13001,16001,19001,21001,24001,28001,51001,54001,55001,61001,69001,70001,76001,81001,88001,90001,93001,96001,97001,102001,103001,109001,114001,115001,121001,123001,124001,126001,129001,
A156656 ,1,-17,-18,6749,6822,-2441897,-7394058,14988306853,15126584742,-9866867955647,-49788967645098,79396908872686679,240385445052560262,-226519192557338133197,-1600244381647342824138,13919437045277101776460901,14047690539171597748097382,
A156657 ,0,1,2,3,4,6,8,9,10,12,13,14,15,16,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,
@@ -157035,7 +157035,7 @@ A157030 ,1,2,0,2,1,0,4,0,1,0,2,1,1,1,0,8,3,0,0,1,0,2,1,1,1,1,1,0,10,0,5,0,1,0,1,
A157031 ,1,1,3,3,9,3,19,7,21,13,51,7,87,17,39,51,175,11,239,21,169,111,415,15,489,185,313,219,1017,15,1413,283,763,415,981,513,3057,839,1259,497,4425,93,5605,893,1311,2259,7505,521,8267,1429,5473,3311,13821,1449,11135,4095,
A157032 ,1,3,6,8,1,8,8,8,8,8,9,0,8,8,8,0,8,0,8,1,1,0,3,8,3,8,8,0,8,3,3,3,8,8,6,6,9,9,1,1,8,1,8,6,9,8,8,1,3,9,3,1,8,3,1,6,8,3,0,0,8,3,1,1,9,1,3,1,3,1,0,3,1,8,6,8,8,1,1,0,0,8,9,1,9,1,1,9,1,1,0,6,8,8,8,8,6,6,8,9,0,8,0,9,3,
A157033 ,2,11,1009,10000019,1000000000000037,10000000000000000000000000000033,1000000000000000000000000000000000000000000000000000000000000121,
-A157034 ,1,9,19,37,33,121,283,37,241,3259,2823,67017,13989,9523,
+A157034 ,2,1,9,19,37,33,121,283,37,241,3259,2823,67017,13989,9523,
A157035 ,7,97,9973,99999989,9999999999999937,99999999999999999999999999999979,9999999999999999999999999999999999999999999999999999999999999949,
A157036 ,3,3,27,11,63,21,51,17,813,377,7017,27381,7763,1133,
A157037 ,6,10,22,30,34,42,58,66,70,78,82,105,114,118,130,142,154,165,174,182,202,214,222,231,238,246,255,273,274,282,285,286,298,310,318,345,357,358,366,370,382,385,390,394,399,418,430,434,442,454,455,465,474,478,
@@ -158824,9 +158824,9 @@ A158819 ,0,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,2,1,1,1,1,0,0,1,1,1,1,1,2,1
A158820 ,1,-5,-1,35,-227,-1435,27599,123095,-2428187,-18154423,1002748195,4095412475,-11278566075977,-1310672758291,9265563303353,564709713458975,-387913690488413419,-315149059886480239,381102197975912820173,221139612243078051395,
A158821 ,1,1,1,2,0,1,3,0,0,1,4,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,1,7,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,0,0,
A158822 ,1,3,1,6,3,2,10,6,5,3,15,10,9,7,4,21,15,14,12,9,5,28,21,20,18,15,11,6,36,28,27,25,22,18,13,7,45,36,35,33,30,26,21,15,8,55,45,44,42,39,35,30,24,17,9,66,55,54,52,49,45,40,34,27,19,10,
-A158823 ,1,3,1,6,2,2,10,3,4,3,15,4,6,6,4,21,5,8,9,8,5,28,6,10,12,12,10,6,36,7,12,15,16,15,12,7,45,8,14,18,20,20,18,14,8,55,9,16,21,24,25,24,21,16,9,66,10,18,24,28,30,30,28,24,18,10,78,11,20,27,32,35,36,35,32,27,20,11,91,
-A158824 ,1,4,1,10,3,2,20,6,6,3,35,10,12,9,4,56,15,20,18,12,5,84,21,30,30,24,15,6,120,28,42,42,40,30,18,7,165,36,56,63,60,50,36,21,8,220,45,72,84,84,75,60,42,24,9,286,55,90,108,112,105,90,70,48,27,10,
-A158825 ,1,1,1,1,2,2,1,3,6,5,1,4,12,21,14,1,5,20,54,80,42,1,6,30,110,260,322,132,1,7,42,195,640,1310,1348,429,1,8,56,315,1330,3870,6824,5814,1430,1,9,72,476,2464,9380,24084,36478,25674,4862,1,10,90,684,4200,19852,67844,
+A158823 ,1,3,1,6,2,2,10,3,4,3,15,4,6,6,4,21,5,8,9,8,5,28,6,10,12,12,10,6,36,7,12,15,16,15,12,7,45,8,14,18,20,20,18,14,8,55,9,16,21,24,25,24,21,16,9,66,10,18,24,28,30,30,28,24,18,10,78,11,20,27,32,35,36,35,32,27,20,11,
+A158824 ,1,4,1,10,3,2,20,6,6,3,35,10,12,9,4,56,15,20,18,12,5,84,21,30,30,24,15,6,120,28,42,45,40,30,18,7,165,36,56,63,60,50,36,21,8,220,45,72,84,84,75,60,42,24,9,286,55,90,108,112,105,90,70,48,27,10,364,66,110,135,144,140,126,105,80,54,30,11,
+A158825 ,1,1,1,1,2,2,1,3,6,5,1,4,12,21,14,1,5,20,54,80,42,1,6,30,110,260,322,132,1,7,42,195,640,1310,1348,429,1,8,56,315,1330,3870,6824,5814,1430,1,9,72,476,2464,9380,24084,36478,25674,4862,1,10,90,684,4200,19852,67844,153306,199094,115566,16796,
A158826 ,1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,35520498,204773400,1191572004,6990859416,41313818217,245735825082,1470125583756,8840948601024,53417237877396,324123222435804,1974317194619712,
A158827 ,1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,294469974,2006018748,13784115468,95444016984,665407010349,4667570034444,32922870719664,233389493503968,1662048903052380,11885333877149532,
A158828 ,1,5,30,195,1330,9380,67844,500619,3755156,28558484,219767968,1708590960,13403300208,105983648060,844009565176,6764300053390,54525119251104,441811163402124,3597005618194848,29412560840221272,
@@ -160504,7 +160504,7 @@ A160499 ,1,0,1,1,3,0,1,2,0,0,1,1,3,0,3,4,3,0,1,3,1,0,1,2,0,0,0,1,3,0,1,0,1,0,3,0
A160500 ,691,1399,1699,5791,6091,6691,6793,7297,8599,10993,12391,12799,13999,14197,14293,15091,15391,15991,17599,18493,18691,19699,22699,22993,23899,24499,24799,25693,26893,27397,28099,28297,28393,29191,33493,
A160501 ,9,251,16627,48844509,13109522141,232643574681223,144347818589843079,8863082234840576951801,100000008862938119652501095929,192043424957750480504146841291811,
A160502 ,1,4,6,2,5,9,0,7,3,5,0,4,4,3,6,4,6,9,9,5,4,6,1,4,5,4,4,6,7,2,0,5,3,4,6,2,1,0,7,4,7,4,4,8,6,4,7,4,8,8,2,1,1,0,9,3,6,4,2,0,0,6,2,4,3,5,4,5,2,2,9,4,3,7,8,5,8,8,1,5,0,3,5,5,2,1,9,2,9,2,2,1,5,9,2,4,0,8,9,2,3,6,9,7,5,
-A160503 ,2,5,2,1229,2,2,664579,3,2,97,19,2,11,2,19,5,3,2,23,2,2,2,73,2,2,7,3,
+A160503 ,2,5,2,1229,2,2,664579,3,2,97,19,2,11,2,19,5,3,2,23,2,2,2,73,2,2,7,3,2,
A160504 ,1,1,1,3,6,6,6,15,15,18,18,18,21,21,21,21,27,27,29,38,38,47,59,59,72,72,72,84,90,90,96,96,97,109,109,112,123,123,123,141,141,143,153,153,161,167,167,170,181,181,186,186,186,193,194,194,202,202,202,210,216,216,
A160505 ,1,3,6,18,36,180,360,1080,2160,6480,12960,64800,129600,388800,777600,2332800,4665600,23328000,46656000,139968000,279936000,839808000,1679616000,8398080000,16796160000,50388480000,100776960000,
A160506 ,1,5,20,65,190,502,1245,2910,6505,13965,29005,58455,114810,220240,413775,762635,1381550,2463060,4327445,7500260,12836645,21712470,36323930,60143320,98620425,160238035,258110955,412367705,653709340,1028658150,1607306688,
@@ -162452,7 +162452,7 @@ A162447 ,12,720,181440,7257600,399168000,1307674368000,73229764608000,7316998520
A162448 ,-11,863,-215641,41208059,-9038561117,28141689013943,-2360298440602051,3420015713873670001,-147239749512798268300237,176556159649301309969405807,-178564975300377173768513546347,
A162449 ,18,2700,992250,250047000,68075295750,253103949598500,24677635085853750,40753351656067050000,1969563638017107451068750,2619519638562752909921437500,2910024366479362207631724918750,
A162450 ,125673984,132978564,136925784,136978425,138572964,139876254,139876425,143297856,143857296,145827396,146385729,148567329,149572836,149872356,153728964,153762894,153764289,158273964,158763249,158769324,162573984,
-A162451 ,24,111,126,128,137,147,248,426,488,575,664,834,3060,4016,4464,4717,5025,5484,6036,7049,8064,8256,8704,8772,9081,32768,38463,57399,59177,78741,86964,94987,
+A162451 ,24,111,126,128,137,147,248,426,488,575,664,834,2317,3060,3968,4016,4464,4717,5025,5484,6036,7049,8064,8256,8704,8772,8919,8946,8973,9081,10535,10575,12943,13467,22553,23478,28082,28488,30927,32768,36864,38463,41664,48657,57399,59177,78078,78741,81075,86964,94987,
A162452 ,1,46,1080,17250,210794,2101418,17796503,131648504,868101374,5182032940,28344317261,143450494506,677150551521,3001361428036,12561988338047,49889607533966,188796675237026,683282982630926,2372613717733406,
A162453 ,1,1,2,1,2,3,1,5,3,4,1,5,9,4,5,1,9,15,12,5,6,1,9,24,24,15,6,7,1,14,36,46,30,18,7,8,1,14,58,70,65,36,21,8,9,1,20,76,130,110,78,42,24,9,10,1,20,111,196,200,144,91,48,27,10,11,1,27,150,314,335,273,168,104,54,30,
A162454 ,7,43,901830931,
@@ -165230,7 +165230,7 @@ A165225 ,1,5,45,425,4025,38125,361125,3420625,32400625,306903125,2907028125,2753
A165226 ,0,1,5,1,31,1,41,1,31,1,61,1,3421,1,-1,1,4127,1,-43069,1,174941,1,-854375,1,236366821,1,-8553097,1,23749461899,1,-8615841261683,1,7709321041727,1,-2577687858361,1,26315271553055396563,1,-2929993913841553,1,
A165227 ,0,1,0,6,5,9,3,5,7,6,6,7,5,0,9,7,7,8,9,3,0,7,7,8,4,4,9,0,6,5,7,8,5,4,2,9,9,4,5,7,4,7,7,5,4,6,4,7,7,4,9,2,1,4,4,3,4,0,4,4,0,6,4,6,8,5,9,3,0,0,1,5,3,7,6,5,9,8,4,1,8,1,2,1,3,5,8,8,0,1,0,7,3,2,5,1,2,1,6,7,5,6,8,0,7,
A165228 ,14,19,16,16,22,12,11,13,16,10,22,24,15,15,21,16,23,20,22,17,11,20,14,18,19,19,13,15,21,20,14,16,12,26,18,16,14,13,16,19,15,16,23,15,14,20,12,12,39,27,16,17,14,40,19,18,19,17,14,22,12,38,19,20,16,21,21,19,23,
-A165229 ,1,6,66,756,8676,99576,1142856,13116816,150544656,1727834976,19830751776,
+A165229 ,1,6,66,756,8676,99576,1142856,13116816,150544656,1727834976,19830751776,227602011456,2612239626816,29981263453056,344101723675776,3949333103390976,45327386898637056,520232644163298816,5970827408567763456,68528533037833368576,786517432002593842176,
A165230 ,1,7,91,1225,16513,222607,3000907,40454449,545355937,7351801975,99107736091,1336045691449,18010885527649,242800077546943,3273124886963659,44124147874662625,594826196036531137,8018697709388797543,108097984559187447643,1457240899862902684201,
A165231 ,1,8,120,1856,28736,444928,6888960,106663936,1651511296,25570869248,395921817600,6130182127616,94915539501056,1469607174995968,22754390483927040,352313390342864896,5454979121614422016,
A165232 ,1,9,153,2673,46737,817209,14289129,249849441,4368687777,76387735017,1335661040313,23354409110481,408358414625841,7140261781270809,124849486331241993,
@@ -165961,7 +165961,7 @@ A165956 ,1,1,2,2,4,4,4,2,8,8,5,6,9,7,7,4,11,11,10,10,12,8,9,10,14,13,11,11,14,12
A165957 ,1,4,6,8,9,0,2,4,5,6,8,0,2,4,8,10,12,14,16,0,6,9,12,15,18,24,27,0,8,16,20,24,32,36,0,5,10,20,25,30,35,40,0,12,18,24,30,36,48,54,0,14,28,35,42,49,56,0,8,16,32,40,48,56,64,0,9,18,27,36,45,54,72,81,0,0,0,0,0,0,0,1,2,
A165958 ,0,3,2,1,6,5,4,9,8,7,
A165959 ,2,3,5,5,5,11,3,7,3,9,5,11,7,9,7,11,15,13,27,25,21,15,13,11,5,17,7,3,11,9,15,9,21,13,3,15,13,7,5,15,11,11,17,15,27,21,15,13,7,21,19,15,9,3,17,15,7,7,7,9,9,17,15,11,9,5,5,21,17,11,7,15,9,
-A165960 ,3,20,100,612,4389,35688,325395,3288490,36489992,441093864,5770007009,81213878830,
+A165960 ,1,1,2,3,20,100,612,4389,35688,325395,3288490,36489992,441093864,5770007009,81213878830,1223895060315,19662509071056,335472890422812,6057979285535388,115434096553014565,2314691409652237700,48723117262650147387,1074208020519710570054,
A165961 ,1,5,20,102,627,4461,36155,328849,3317272,36757822,443846693,5800991345,81593004021,1228906816941,19733699436636,336554404751966,6075478765948135,115734570482611885,2320148441078578447,48827637296350480457,1076313671861962141616,
A165962 ,1,5,18,95,600,4307,35168,321609,3257109,36199762,438126986,5736774126,80808984725,1218563180295,19587031966352,334329804347219,6039535339644630,115118210694558105,2308967760171049528,48613722701436777455,1072008447320752890459,
A165963 ,0,16,80,516,3794,31456,290970,2974380,33311520,405773448,5342413414,75612301688,
@@ -165991,7 +165991,7 @@ A165986 ,6,14,26,38,74,86,134,158,194,206,218,254,326,386,446,458,554,614,626,69
A165987 ,1099258818702,8792791182238,29674231047422,70337212371066,
A165988 ,0,3,12,9,24,15,36,21,48,27,60,33,72,39,84,45,96,51,108,57,120,63,132,69,144,75,156,81,168,87,180,93,192,99,204,105,216,111,228,117,240,123,252,129,264,135,276,141,288,147,300,153,312,159,324,165,336,171,348,177,
A165989 ,534,659,1727,1852,2920,3045,4113,4238,5306,5431,6499,6624,7692,7817,8885,9010,10078,10203,11271,11396,12464,12589,13657,13782,14850,14975,16043,16168,17236,17361,18429,18554,19622,19747,20815,20940,22008,22133,23201,
-A165990 ,3,7,12,15,99,188,
+A165990 ,3,7,12,15,99,188,843,1567,1388,12823,25739,24828,203347,169975,1793132,3247295,3281747,33047100,46475931,223888367,464656140,443782407,3392754203,6320720892,28126943139,51929697511,46812642508,430604078639,875439722435,832171221180,
A165991 ,0,5,8,9,10,11,13,14,17,19,20,21,22,23,24,25,26,27,29,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,56,57,58,59,60,61,62,63,64,65,66,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,86,87,
A165992 ,1,3,3,9,12,12,27,39,51,51,81,120,171,222,222,243,363,534,756,978,978,729,1092,1626,2382,3360,4338,4338,2187,3279,4905,7287,10647,14985,19323,19323,6561,9840,14745,22032,32679,47664,66987,86310,86310,19683,
A165993 ,0,1,4,11,31,44,80,103,157,252,293,420,520,575,695,884,1105,1180,1431,1617,1704,2007,2217,2552,3040,3300,3439,3713,3852,4144,5255,5595,6120,6305,7252,7457,8060,8695,9141,9804,10507,10740,11983,12224,12740,
@@ -167052,7 +167052,7 @@ A167047 ,1,3,5,7,9,2,11,13,4,15,17,6,19,21,8,23,25,27,10,29,31,12,33,35,37,14,39
A167048 ,1,18,306,5202,88434,1503378,25557426,434476242,7386096114,125563633938,2134581776946,36287890208082,616894133537394,10487200270135545,178282404592301664,3030800878069084224,51523614927173682720,
A167049 ,1,19,342,6156,110808,1994544,35901792,646232256,11632180608,209379250944,3768826516992,67838877305856,1221099791505408,21979796247097173,395636332447746036,7121453984059373415,
A167050 ,2,3,5,7,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,62,65,69,74,77,82,85,86,87,91,93,94,95,102,105,110,114,130,138,154,165,170,174,182,186,190,195,222,230,231,238,246,255,258,266,273,282,285,286,290,310,
-A167051 ,1,2,4,7,8,10,25,26,28,79,80,82,241,242,244,727,728,730,2185,
+A167051 ,1,2,4,7,8,10,25,26,28,79,80,82,241,242,244,727,728,730,2185,2186,2188,6559,6560,6562,19681,19682,19684,59047,59048,59050,177145,177146,177148,531439,531440,531442,1594321,1594322,1594324,4782967,4782968,4782970,14348905,
A167052 ,6,3,4,4,8,8,3,4,9,8,13,6,7,7,11,11,9,10,15,14,8,10,11,11,15,15,10,11,16,15,6,13,14,14,18,18,13,14,19,18,10,17,18,18,22,22,17,18,23,22,12,19,20,20,24,24,19,20,25,24,7,14,15,15,19,19,14,15,20,19,7,14,15,15,19,19,
A167053 ,3,19,39,81,165,333,335,673,1347,1349,1351,1353,1355,1357,1359,2721,2723,2725,2727,5457,5459,5461,5463,5465,5467,5469,10941,10943,10945,10947,21897,21899,21901,21903,21905,21907,21909,43821,43823,43825,43827,43829,43831,
A167054 ,15,19,41,83,167,337,673,1361,2729,5471,10949,21911,43853,87719,175447,350899,701819,1403641,2807303,5614657,11229331,22458671,44917381,89834777,179669557,359339171,718678369,
@@ -171844,8 +171844,8 @@ A171839 ,1,0,0,1,0,0,1,1,0,0,3,2,1,0,0,6,8,3,1,0,0,15,22,15,4,1,0,0,36,68,52,24,
A171840 ,1,1,2,1,2,5,1,2,4,15,1,2,4,9,52,1,2,4,8,23,203,1,2,4,8,17,65,877,1,2,4,8,16,40,199,4140,1,2,4,8,16,33,104,654,21147,1,2,4,8,16,32,73,291,2296,115975,1,2,4,8,16,32,65,177,857,8569,678570,
A171841 ,1,3,8,22,68,241,974,4410,21969,118698,688301,4250788,
A171842 ,1,1,2,4,9,21,50,120,289,697,1682,4060,9801,23661,57122,137904,332929,803761,1940450,4684660,11309769,27304197,65918162,159140520,384199201,927538921,2239277042,5406093004,13051463049,31509019101,76069501250,183648021600,443365544449,1070379110497,
-A171843 ,1,1,3,1,3,8,1,3,6,21,1,3,6,12,55,1,3,6,10,24,144,1,3,6,10,17,48,377,1,3,6,10,15,30,96,987,1,3,6,10,15,23,53,192,2584,1,3,6,10,15,21,57,93,384,6765,
-A171844 ,1,4,12,31,77,188,462,1148,2887,7355,18789,48419,125291,
+A171843 ,1,1,3,1,3,8,1,3,6,21,1,3,6,12,55,1,3,6,10,24,144,1,3,6,10,17,48,377,1,3,6,10,15,30,96,987,1,3,6,10,15,23,53,192,2584,1,3,6,10,15,21,37,93,384,6765,1,3,6,10,15,21,30,61,163,768,17711,1,3,6,10,15,21,28,45,100,286,1536,46368,
+A171844 ,1,4,12,31,77,188,462,1148,2887,7335,18789,48419,125321,325381,846713,2206891,5758797,15040102,39304237,102760572,268757551,703079117,1839625401,4814107671,12599351527,32977310272,86319400527,225954695164,591492569038,1548419254590,
A171845 ,0,4,6,9,12,15,18,21,25,26,27,30,33,34,35,39,42,45,49,50,51,55,56,57,60,63,64,65,69,72,75,76,77,81,85,86,87,91,92,93,94,95,99,102,105,108,111,115,116,117,118,119,120,121,122,123,124,125,129,133,134,135,138,
A171846 ,1,1,1,1,0,1,1,0,2,1,1,0,3,2,1,0,1,1,0,4,3,3,1,2,2,1,1,0,5,4,6,4,4,4,5,2,1,0,1,1,0,6,5,10,9,9,7,11,8,5,3,3,2,2,1,1,0,7,6,15,16,18,14,20,20,16,10,11,8,8,6,5,2,1,0,1,1,0,8,7,21,25,32,28,36,39,41,29,27,24,25,20,17,
A171847 ,0,0,0,2,7,22,68,198,563,1578,4367,11980,32648,88500,238886,642598,1723629,4612170,12316357,32832302,87390763,232305470,616812557,1636084020,4335770052,11480937084,30379110906,80332372838,212300488377,
@@ -173080,7 +173080,7 @@ A173075 ,1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,12,12,5,1,1,6,18,25,18,6,1,1,7,25,44,
A173076 ,1,1,1,1,3,1,1,4,4,1,1,7,13,7,1,1,8,21,21,8,1,1,13,46,67,46,13,1,1,14,60,114,114,60,14,1,1,23,123,295,389,295,123,23,1,1,24,147,419,685,685,419,147,24,1,1,41,300,1015,2001,2491,2001,1015,300,41,1,
A173077 ,1,1,1,1,4,1,1,5,5,1,1,12,23,12,1,1,13,36,36,13,1,1,32,122,181,122,32,1,1,33,155,304,304,155,33,1,1,88,513,1270,1689,1270,513,88,1,1,89,602,1784,2960,2960,1784,602,89,1,1,252,1988,6923,13817,17261,13817,6923,1988,252,1,
A173078 ,1,3,11,23,51,103,211,423,851,1703,3411,6823,13651,27303,54611,109223,218451,436903,873811,1747623,3495251,6990503,13981011,27962023,55924051,111848103,223696211,447392423,894784851,1789569703,3579139411,
-A173079 ,2,3,12,15,17,22,35,124,191,774,1405,1522,3988,6220,7448,8038,11404,63027,161153,
+A173079 ,1,2,3,12,15,17,22,35,124,191,774,1405,1522,3988,6220,7448,8038,11404,63027,161153,
A173080 ,8,24,40,56,72,88,104,120,128,136,152,168,179,184,200,216,232,248,264,280,296,312,323,328,344,358,360,376,384,389,392,398,408,424,437,440,456,459,472,488,493,504,520,536,537,552,568,569,584,600,616,621,632,
A173081 ,0,6,28,167,964,6305,45082,335919,2605867,20841010,170395131,
A173082 ,6,51,65,115,133,141,159,187,201,209,213,287,291,295,327,339,361,407,411,413,471,493,511,519,537,559,579,597,633,649,687,695,723,799,813,831,835,871,917,939,1007,1041,1047,1079,1135,1167,1189,1195,1199,1227,
@@ -173349,11 +173349,11 @@ A173344 ,0,1,0,-2,-3,0,8,13,0,-34,-55,0,144,233,0,-610,-987,0,2584,4181,0,-10946
A173345 ,0,0,0,0,1,2,3,4,5,7,9,11,13,15,18,21,24,27,30,34,38,42,46,50,56,62,68,74,80,87,94,101,108,115,123,131,139,147,155,164,173,182,191,200,210,220,230,240,250,262,274,286,298,310,323,336,349,362,375,389,403,417,
A173346 ,0,4,16,144,324,625,
A173347 ,21,233,196418,9227465,165580141,2971215073,53316291173,2504730781961,3416454622906707,51680708854858323072,184551825793033096366333,898923707008479989274290850145,3210056809456107725247980776292056,
-A173348 ,12,93,239,4896,4904,6546,7806,9104,20542,35962,43783,96569,616400,635331,842163,7888432,
-A173349 ,892,1110,1498,1827,3657,9249,10637,27590,63500,63508,248461,300221,357450,1317619,4782975,6245380,6376350,7486710,
-A173350 ,3,21,145,1005,1746,5559,29005,34320,76053,146402,154269,553624,853772,853780,1841222,2582634,3051972,
-A173351 ,15,42,71,168,9172,15844,542482,548554,5947459,9825757,
-A173352 ,2,4,23,122,199,408,4995,7320,44217,177682,394826,1706886,1738064,8403388,
+A173348 ,12,93,239,4896,4904,6546,7806,9104,20542,35962,43783,96569,616400,635331,842163,7888432,450177181,
+A173349 ,892,1110,1498,1827,3657,9249,10637,27590,63500,63508,248461,300221,357450,1317619,4782975,6245380,6376350,7486710,10059286,22580324,26040615,34827846,123838550,170331287,178384607,234365487,483178063,
+A173350 ,3,21,145,1005,1746,5559,29005,34320,76053,146402,154269,553624,853772,853780,1841222,2582634,3051972,23630121,41571007,76908458,98649561,106586932,114021359,316366090,633141499,
+A173351 ,15,42,71,168,9172,15844,542482,548554,5947459,9825757,176874995,671960665,
+A173352 ,2,4,23,122,199,408,4995,7320,44217,177682,394826,1706886,1738064,8403388,21194961,110525339,314033376,328840890,
A173353 ,2,3,32,33,34,88,442,498,942,2266144,
A173354 ,97,37840,199652,2905727,
A173355 ,2,3,48,73,436,23494,37381,621706,781913,2351612,
@@ -173882,16 +173882,16 @@ A173877 ,1,3,6,17,13,40,27,106,78,127,79,391,129,321,358,832,285,1070,409,1549,
A173878 ,1,3,7,23,19,65,46,202,156,281,183,972,333,903,1029,2507,912,
A173879 ,5,53,389,509,593,599,839,2879,2963,4013,4799,5273,6473,6719,6869,7499,8243,10589,11003,11069,15959,17483,20123,21383,25073,25583,27059,28319,32213,34019,34253,34913,37013,38453,38609,38933,41039,42569,43283,
A173880 ,7,61,151,157,571,997,1447,1831,2251,3121,4057,4177,5011,5737,6907,10321,10357,10567,11941,15601,16477,19267,19597,20347,22447,22531,23131,24121,24337,29587,29641,30181,30817,33577,37201,37447,38671,
-A173881 ,1,1,1,1,2,1,1,6,6,1,1,12,36,12,1,1,20,120,120,20,1,1,30,300,600,300,30,1,1,42,630,2100,2100,630,42,1,1,56,1176,5880,9800,5880,1176,56,1,1,72,2016,14112,35280,35280,14112,2016,72,1,1,90,3240,30240,105840,158760,
-A173882 ,1,1,1,1,6,1,1,24,24,1,1,60,240,60,1,1,120,1200,1200,120,1,1,210,4200,10500,4200,210,1,1,336,11760,58800,58800,11760,336,1,1,504,28224,246960,493920,246960,28224,504,1,1,720,60480,846720,2963520,2963520,
+A173881 ,1,1,1,1,2,1,1,6,6,1,1,12,36,12,1,1,20,120,120,20,1,1,30,300,600,300,30,1,1,42,630,2100,2100,630,42,1,1,56,1176,5880,9800,5880,1176,56,1,1,72,2016,14112,35280,35280,14112,2016,72,1,1,90,3240,30240,105840,158760,105840,30240,3240,90,1,
+A173882 ,1,1,1,1,6,1,1,24,24,1,1,60,240,60,1,1,120,1200,1200,120,1,1,210,4200,10500,4200,210,1,1,336,11760,58800,58800,11760,336,1,1,504,28224,246960,493920,246960,28224,504,1,1,720,60480,846720,2963520,2963520,846720,60480,720,1,
A173883 ,0,0,0,0,0,1,1,2,2,2,3,2,2,3,2,2,2,4,4,4,4,4,4,4,4,4,4,3,4,4,4,4,5,4,4,4,4,4,4,4,4,4,4,4,4,4,8,8,8,8,6,5,4,4,4,4,4,6,4,5,4,5,4,5,5,4,4,4,4,5,6,5,4,6,4,4,4,4,4,4,4,8,8,8,8,8,8,8,
A173884 ,1,1,1,1,3,1,1,21,21,1,1,105,735,105,1,1,465,16275,16275,465,1,1,1953,302715,1513575,302715,1953,1,1,8001,5208651,115334415,115334415,5208651,8001,1,1,32385,86370795,8032483935,35572428855,8032483935,
A173885 ,1,1,1,1,16,1,1,208,208,1,1,2080,27040,2080,1,1,19360,2516800,2516800,19360,1,1,176176,213172960,2131729600,213172960,176176,1,1,1591408,17522993488,1630986316960,1630986316960,17522993488,1591408,1,1,
A173886 ,1,1,1,1,1,1,1,2,2,1,1,6,12,6,1,1,15,90,90,15,1,1,40,600,1800,600,40,1,1,104,4160,31200,31200,4160,104,1,1,273,28392,567840,1419600,567840,28392,273,1,1,714,194922,10135944,67572960,67572960,10135944,194922,
A173887 ,1,1,1,1,2,1,1,6,6,1,1,30,90,30,1,1,120,1800,1800,120,1,1,520,31200,156000,31200,520,1,1,2184,567840,11356800,11356800,567840,2184,1,1,9282,10135944,878448480,3513793920,878448480,10135944,9282,1,1,39270,
A173888 ,0,1,4,5,6,7,8,9,10,17,19,23,25,32,51,55,65,87,129,132,159,171,175,180,242,315,324,358,393,435,467,491,501,507,555,591,680,786,800,1070,1459,1650,1707,2813,2923,3281,4217,5153,6287,6365,6462,10088,10367,14289,
-A173889 ,1,1,1,1,2,1,1,12,12,1,1,120,720,120,1,1,360,21600,21600,360,1,1,840,151200,1512000,151200,840,1,1,1680,705600,21168000,21168000,705600,1680,1,1,3024,2540160,177811200,533433600,177811200,2540160,3024,1,1,
-A173890 ,1,1,1,1,2,1,1,72,72,1,1,840,30240,840,1,1,2880,1209600,1209600,2880,1,1,7560,10886400,127008000,10886400,7560,1,1,16800,63504000,2540160000,2540160000,63504000,16800,1,1,33264,279417600,29338848000,100590336000,
+A173889 ,1,1,1,1,2,1,1,12,12,1,1,120,720,120,1,1,360,21600,21600,360,1,1,840,151200,1512000,151200,840,1,1,1680,705600,21168000,21168000,705600,1680,1,1,3024,2540160,177811200,533433600,177811200,2540160,3024,1,
+A173890 ,1,1,1,1,2,1,1,72,72,1,1,840,30240,840,1,1,2880,1209600,1209600,2880,1,1,7560,10886400,127008000,10886400,7560,1,1,16800,63504000,2540160000,2540160000,63504000,16800,1,1,33264,279417600,29338848000,100590336000,29338848000,279417600,33264,1,
A173891 ,1,3,16,37,40,47,55,56,74,103,108,111,119,130,161,165,185,188,195,200,219,240,272,273,292,340,359,388,420,427,465,466,509,521,554,600,606,622,630,634,668,683,684,703,710,711,734,762,792,814,822,823,830,831,
A173892 ,257,1747,3307,5107,5387,6317,6367,12647,13457,14747,15797,15907,17477,18217,19477,23327,26177,30097,30637,53617,56087,62207,63697,71347,75527,77557,78797,80917,82787,83437,84437,89107,89387,91297,94427,95267,
A173893 ,26,175,331,511,539,632,637,1265,1346,1475,1580,1591,1748,1822,1948,2333,2618,3010,3064,5362,5609,6221,6370,7135,7553,7756,7880,8092,8279,8344,8444,8911,8939,9130,9443,9527,
@@ -174044,8 +174044,8 @@ A174039 ,1,1,1,1,12,1,1,37,37,1,1,116,452,116,1,1,305,2544,2544,305,1,1,752,1249
A174040 ,1,1,1,1,19,1,1,71,71,1,1,281,2064,281,1,1,942,19925,19925,942,1,1,3000,182094,372304,182094,3000,1,1,9265,708780,2645662,2645662,708780,9265,1,1,28435,2470768,15065554,26673373,15065554,2470768,28435,1,1,86456,
A174041 ,8,9,14,15,20,21,26,27,32,33,38,39,44,45,50,51,56,57,62,63,68,69,74,75,80,81,86,87,92,93,98,99,104,105,110,111,116,117,122,123,128,129,134,135,140,141,146,147,152,153,158,159,164,165,170,171,176,177,182,183,
A174042 ,1,9,10,24,28,67,195,361,362,382,459,462,470,759,765,766,794,864,869,909,1189,1300,1303,1374,1378,1642,1657,1659,3727,3755,4187,4368,4413,4677,4684,4721,4927,4945,5221,5270,5313,5409,5627,5945,7587,7588,7789,
-A174043 ,1,1,1,1,4,1,1,10,10,1,1,18,38,18,1,1,26,97,97,26,1,1,39,206,344,206,39,1,1,53,389,974,974,389,53,1,1,70,669,2348,3522,2348,669,70,1,1,86,1076,5033,10575,10575,5033,1076,86,1,
-A174044 ,1,1,1,1,5,1,1,16,16,1,1,36,92,36,1,1,49,276,276,49,1,1,93,673,1265,673,93,1,1,124,1484,4004,4004,1484,124,1,1,204,2832,12400,18060,12400,2832,204,1,1,237,5244,26416,57580,57580,26416,5244,237,1,
+A174043 ,1,1,1,1,4,1,1,10,10,1,1,18,38,18,1,1,26,97,97,26,1,1,39,206,344,206,39,1,1,53,389,974,974,389,53,1,1,70,669,2348,3522,2348,669,70,1,1,86,1076,5033,10575,10575,5033,1076,86,1,1,105,1644,9890,27704,38784,27704,9890,1644,105,1,
+A174044 ,1,1,1,1,5,1,1,16,16,1,1,36,92,36,1,1,49,276,276,49,1,1,93,673,1265,673,93,1,1,124,1484,4004,4004,1484,124,1,1,204,2832,12400,18060,12400,2832,204,1,1,237,5244,26416,57580,57580,26416,5244,237,1,1,289,7729,53024,151756,211692,151756,53024,7729,289,1,
A174045 ,1,1,1,1,6,1,1,24,24,1,1,70,230,70,1,1,90,881,881,90,1,1,231,2790,7060,2790,231,1,1,295,8383,28270,28270,8383,295,1,1,684,21441,181680,242172,181680,21441,684,1,1,750,58320,378009,882549,882549,378009,58320,750,1,
A174046 ,2,3,4,6,14,16,29,356,358,359,403,446,464,485,652,655,764,861,866,1123,1301,1304,1324,1328,1358,1486,1610,2631,2632,3735,3931,3953,3956,3957,4679,4855,4931,5222,5226,5269,5283,5292,5403,5427,5445,
A174047 ,1,4,5,7,8,10,11,12,14,16,18,19,20,22,23,24,26,27,29,31,33,34,35,37,39,40,41,42,44,45,48,49,50,52,53,55,56,57,63,64,65,66,68,70,74,76,78,79,81,82,83,84,86,87,89,91,95,97,98,100,105,106,111,112,113,115,116,117,119,121,125,126,128,129,131,
@@ -174073,15 +174073,15 @@ A174068 ,1,1,2,2,4,5,7,9,13,17,23,29,38,48,62,77,98,121,153,187,233,283,349,422,
A174069 ,5,13,14,25,29,30,41,50,54,55,61,77,85,86,90,91,110,113,126,135,139,140,145,149,174,181,190,194,199,203,204,221,230,245,255,265,271,280,284,285,294,302,313,330,355,365,366,371,380,384,385,415,421,434,446,451,
A174070 ,14,29,30,50,54,55,77,86,90,91,110,126,135,139,140,149,174,190,194,199,203,204,230,245,255,271,280,284,285,294,302,330,355,365,366,371,380,384,385,415,434,446,451,476,492,501,505,506,509,510,534,559,590,595,
A174071 ,30,54,55,86,90,91,126,135,139,140,174,190,199,203,204,230,255,271,280,284,285,294,330,355,366,371,380,384,385,415,446,451,476,492,501,505,506,510,534,559,595,615,620,630,636,645,649,650,679,728,730,734,764,
-A174072 ,114,674,4714,37754,340404,
-A174073 ,100,594,4389,35744,325395,
-A174074 ,22,109,657,4625,37186,336336,3379058,37328103,449669577,5866178493,82387080624,
-A174075 ,18,93,600,4320,35186,
-A174076 ,108,632,4408,35336,319056,
-A174077 ,80,504,3794,31616,290970,
+A174072 ,1,1,2,6,24,114,674,4714,37754,340404,3412176,37631268,452745470,5900431012,82802497682,1244815252434,19958707407096,339960096280062,6130407887839754,116675071758609742,
+A174073 ,1,1,2,3,24,100,594,4389,35744,325395,3288600,36489992,441091944,5770007009,81213883898,1223895060315,19662509172096,335472890422812,
+A174074 ,2,6,22,109,657,4625,37186,336336,3379058,37328103,449669577,5866178493,82387080624,
+A174075 ,1,6,18,93,600,4320,35168,321630,3257109,36199458,438126986,5736774869,80808984725,1218563192160,19587031966352,334329804180135,6039535339644630,115118210695441900,2308967760171049528,48613722701440862328,1072008447320752890459,
+A174076 ,1,1,2,6,24,108,632,4408,35336,319056,3205824,35451984,427683560,5588310904,78615281768,1184587864512,19033796498496,324852522308160,5868833343451592,111889157407344424,
+A174077 ,1,1,2,0,24,80,504,3794,31616,290970,2973600,33311520,405781344,5342413414,75612197528,1144942063230,18471128518656,316309310084728,5730646943736936,
A174078 ,20,100,600,4244,34264,311424,3143912,34833964,420917638,5513592091,77715460917,
A174079 ,12,84,494,3696,30574,
-A174080 ,21,100,597,4113,32842,292379,
+A174080 ,1,1,2,5,21,100,597,4113,32842,292379,2925367,31983248,383514347,4966286235,69508102006,1039315462467,16627618496319,282023014602100,5075216962675445,96263599713301975,1925002914124917950,
A174081 ,16,40,300,1764,17056,118908,
A174082 ,5,18,91,544,3842,30573,
A174083 ,4,0,40,168,1652,9408,
@@ -174151,8 +174151,8 @@ A174146 ,1,2,5,13,40,131,481,1857,7600,32235,141203,633383,2899885,13498337,6373
A174147 ,0,2,4,2,8,5,12,2,4,9,20,5,24,13,14,2,32,5,36,9,20,21,44,5,8,25,4,13,56,15,60,2,32,33,34,5,72,37,38,9,80,21,84,21,14,45,92,5,12,9,50,25,104,5,54,13,56,57,116,15,120,61,20,2,64,33,132,33,68,35,140,5,144,73,14,
A174148 ,1,1,1,1,12,1,1,42,42,1,1,100,360,100,1,1,195,1700,1700,195,1,1,336,5775,14000,5775,336,1,1,532,15876,75950,75950,15876,532,1,1,792,37632,312816,617400,312816,37632,792,1,1,1125,79920,1058400,3630312,3630312,
A174149 ,1,1,1,1,0,1,1,120,120,1,1,720,5400,720,1,1,2520,84700,84700,2520,1,1,6720,712950,3136000,712950,6720,1,1,15120,4064256,56080500,56080500,4064256,15120,1,1,30240,17745840,619178112,1944810000,619178112,
-A174150 ,1,1,1,1,6,1,1,12,12,1,1,30,60,30,1,1,60,300,300,60,1,1,105,1050,2625,1050,105,1,1,168,2940,14700,14700,2940,168,1,1,252,7056,61740,123480,61740,7056,252,1,1,360,15120,211680,740880,740880,211680,15120,360,1,1,
-A174151 ,1,1,1,1,12,1,1,60,60,1,1,180,900,180,1,1,420,6300,6300,420,1,1,840,29400,88200,29400,840,1,1,1512,105840,740880,740880,105840,1512,1,1,2520,317520,4445280,10372320,4445280,317520,2520,1,1,3960,831600,
+A174150 ,1,1,1,1,6,1,1,12,12,1,1,30,60,30,1,1,60,300,300,60,1,1,105,1050,2625,1050,105,1,1,168,2940,14700,14700,2940,168,1,1,252,7056,61740,123480,61740,7056,252,1,1,360,15120,211680,740880,740880,211680,15120,360,1,
+A174151 ,1,1,1,1,12,1,1,60,60,1,1,180,900,180,1,1,420,6300,6300,420,1,1,840,29400,88200,29400,840,1,1,1512,105840,740880,740880,105840,1512,1,1,2520,317520,4445280,10372320,4445280,317520,2520,1,1,3960,831600,20956320,97796160,97796160,20956320,831600,3960,1,
A174152 ,13,19,43,79,139,151,211,271,373,433,523,643,739,751,769,853,919,1033,1051,1093,1129,1171,1423,1429,1471,1531,1579,1663,1741,1759,1789,1933,2053,2281,2389,2521,2689,2731,2749,2833,3061,3109,3163,3271,3313,3319,
A174153 ,2,8,10,64,76,118,120,258,303,332,364,528,811,1270,1362,1607,2091,2572,3596,8190,
A174154 ,448,30319,976640,21137959,357365350,5109144543,64737165162,749160010737,8080813574550,82425144219429,803491953235264,7545414941610145,68680800264413920,608889093898882615,5278006575696293456,44873569636443901967,375159494582050088590,3090799708762482416287,
@@ -176529,7 +176529,7 @@ A176524 ,1,5,3,2,9,7,0,9,7,1,6,7,5,5,8,9,1,6,5,6,5,5,3,6,8,1,9,9,1,5,7,2,0,4,8,7
A176525 ,6,8,10,12,14,15,18,20,21,22,26,27,28,32,33,34,35,36,38,39,44,45,46,48,50,51,52,55,57,58,62,63,64,65,68,69,74,75,76,77,80,82,85,86,87,91,92,93,94,95,98,99,100,106,111,112,115,116,117,118,119,122,
A176526 ,1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,28,30,35,36,40,42,45,48,50,60,70,72,75,80,84,90,100,105,120,140,144,150,180,200,210,240,300,360,420,600,720,
A176527 ,1,2,5,21,166,2277,49901,1675904,84239935,6231045077,668949067432,103005162942955,22511886374045653,6918461813753405930,2965189776573865320121,1759287329824925168339697,1435531006280642249195752862,1601571709194974043628781397985,2430449338115875591262479128994073,
-A176528 ,1,1,2,2,4,6,8,8,12,20,26,36,44,56,64,64,76,108,128,200,226,286,322,432,476,572,628,
+A176528 ,1,1,2,2,4,6,8,8,12,20,26,36,44,56,64,64,76,108,128,200,226,286,322,432,476,572,628,784,848,960,1024,1024,1100,1292,1400,1944,2072,2432,2632,4000,4226,4746,5032,6292,6614,7406,7838,10368,10844,11900,12472,14872,
A176529 ,1,0,4,7,7,2,2,5,5,7,5,0,5,1,6,6,1,1,3,4,5,6,9,6,9,7,8,2,8,0,0,8,0,2,1,3,3,9,5,2,7,4,4,6,9,4,9,9,7,9,8,3,2,5,4,2,2,6,8,9,4,4,4,9,7,3,2,4,9,3,2,7,7,1,2,2,7,2,2,7,3,3,8,0,0,8,5,8,4,3,6,1,6,3,8,7,0,6,2,5,7,6,4,7,2,
A176530 ,1,0,3,2,2,9,0,6,4,7,4,2,2,3,7,7,0,6,6,6,3,5,6,7,4,8,3,9,2,3,2,6,8,7,1,2,8,0,5,2,2,4,4,9,9,0,1,4,5,8,4,7,5,5,5,7,7,4,5,6,1,2,5,4,8,6,8,7,4,6,4,8,4,1,9,5,8,7,8,0,9,7,4,2,8,2,1,8,7,4,3,7,1,4,1,9,4,6,3,9,7,9,1,9,7,
A176531 ,1,0,2,4,4,0,4,4,2,4,0,8,5,0,7,5,7,7,3,4,9,5,7,2,6,7,5,6,8,3,9,9,6,8,7,9,9,2,3,7,6,3,5,9,2,8,8,4,0,7,5,1,9,9,2,4,3,7,8,7,7,8,8,1,7,9,0,0,0,2,9,6,2,7,5,0,5,5,0,3,4,5,7,0,9,6,9,2,6,4,4,4,6,6,5,9,7,2,0,8,9,0,1,9,8,
@@ -177994,7 +177994,7 @@ A177989 ,2,3,4,5,6,8,9,10,11,13,14,18,19,20,21,22,23,24,25,26,27,28,29,30,33,34,
A177990 ,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,
A177991 ,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,3,1,2,1,1,1,3,1,2,1,1,1,4,1,3,1,2,1,1,1,4,1,3,1,2,1,1,1,1,5,1,4,1,3,1,2,1,1,1,5,1,4,1,3,1,2,1,1,1,
A177992 ,1,1,1,1,3,1,1,7,5,1,1,15,6,5,1,1,31,10,16,5,1,1,63,15,42,15,7,1,1,127,21,99,35,29,7,1,1,255,28,219,70,93,28,9,1,1,511,36,466,126,256,84,46,9,1,
-A177993 ,1,1,1,2,3,1,2,4,3,1,3,8,9,5,1,3,9,13,11,5,1,4,15,28,31,20,7,1,4,16,34,46,40,22,7,1,5,24,62,102,110,78,35,9,1,5,25,70,130,166,148,91,37,9,1,
+A177993 ,1,1,1,2,3,1,2,4,3,1,3,8,9,5,1,3,9,13,11,5,1,4,15,28,31,20,7,1,4,16,34,46,40,22,7,1,5,24,62,102,110,78,35,9,1,5,25,70,130,166,148,91,37,9,1,6,35,115,250,376,400,301,157,54,11,1,6,36,125,295,496,610,553,367,174,56,11,1,
A177994 ,1,1,1,2,1,1,2,1,1,1,3,1,2,1,1,3,1,2,1,1,1,4,1,3,1,2,1,1,4,1,3,1,2,1,1,1,5,1,4,1,3,1,2,1,1,5,1,4,1,3,1,2,1,1,1,
A177995 ,1,2,1,2,0,3,2,0,6,5,2,0,6,0,13,2,0,6,0,26,21,2,0,6,0,26,0,55,2,0,6,0,26,0,110,89,2,0,6,0,26,0,110,0,233,2,0,6,0,26,0,110,0,466,377,1,0,6,0,26,0,110,0,466,0,987,
A177996 ,41,29,23,79,354689,1961870762757168078553,47,40427,4093,4441,2543,1033,659,181194015068926422899222020415627,394502321,22742387,36583,569,14747,12641,167,407987015619859919,100493,3679329001,
@@ -178068,7 +178068,7 @@ A178063 ,1,2,4,7,11,17,23,34,44,62,78,98,122,148,168,213,253,291,325,387,433,487
A178064 ,2,1,2,2,1,0,2,2,2,2,4,4,3,3,2,2,3,2,1,2,2,1,5,4,4,3,4,3,3,2,2,2,2,1,3,1,2,2,1,2,2,1,6,5,5,4,4,3,5,5,3,3,2,5,4,4,4,4,2,3,2,5,4,4,3,3,3,4,3,3,2,3,3,2,2,4,2,2,1,2,1,1,6,6,6,5,6,5,3,6,5,4,5,4,4,4,3,5,4,4,2,6,5,5,4,
A178065 ,1,2,2,2,3,4,3,3,3,3,2,2,3,3,4,4,3,4,5,4,4,5,2,3,3,4,3,4,4,5,5,5,5,6,4,6,5,5,6,5,5,6,2,3,3,4,4,5,3,3,5,5,6,3,4,4,4,4,6,5,6,3,4,4,5,5,5,4,5,5,6,5,5,6,6,4,6,6,7,6,7,7,3,3,3,4,3,4,6,3,4,5,4,5,5,5,6,4,5,5,7,3,4,4,5,
A178066 ,59,108959,176459,4040159,5904959,10497659,25401659,26625659,38192459,89302559,105884159,117288959,155750459,156500159,228614459,251856959,306950459,432224159,491508959,508953659,624500159,682776959,934524959,1092963659,1106892959,
-A178067 ,1,5,3,15,11,6,34,27,19,10,65,54,42,29,15,111,95,78,60,41,21,175,153,130,106,81,55,28,260,231,201,170,138,105,71,36,
+A178067 ,1,5,3,15,11,6,34,27,19,10,65,54,42,29,15,111,95,78,60,41,21,175,153,130,106,81,55,28,260,231,201,170,138,105,71,36,369,332,294,255,215,174,132,89,45,505,459,412,364,315,265,214,162,109,55,671,615,558,500,441,381,320,258,195,131,66,
A178068 ,3,5,23,89,173,233,239,1223,1409,1559,2549,2693,3389,3803,4373,4919,9059,10313,16493,17159,20879,20939,22013,24473,25229,31649,32933,34253,34883,37049,38453,39089,40283,41399,43793,44543,49103,49919,50993,
A178069 ,12345679,24691358,49382716,61728395,86419753,98765432,123456790,135802469,160493827,172839506,197530864,209876543,234567901,246913580,271604938,283950617,308641975,320987654,345679012,358024691,382716049,
A178070 ,11,17,41,73,101,137,251,257,271,353,401,449,641,751,1201,1409,1601,3541,4001,4801,5051,9091,10753,15361,16001,19841,21001,21401,24001,25601,27961,37501,40961,43201,60101,62501,65537,69857,76001,76801,160001,162251,163841,307201,453377,524801,544001,670001,952001,976193,980801,
@@ -178127,7 +178127,7 @@ A178122 ,1,1,1,1,8,1,1,27,27,1,1,82,240,82,1,1,245,1700,1700,245,1,1,732,10571,2
A178123 ,1,1,2,5,16,61,269,1337,7354,44155,286397,1990427,14725738,115356349,952592288,8261093885,74994333994,710656444489,7012302313061,71892455879393,764331907463476,8411953721081635,95684448908132498,
A178124 ,1,-1,1,1,-3,1,0,7,-6,1,-5,-13,26,-10,1,25,11,-101,69,-15,1,-105,76,383,-425,150,-21,1,460,-758,-1494,2599,-1310,286,-28,1,-2315,5536,6215,-16761,11129,-3325,497,-36,1,13935,-40769,-27989,118079,-97272,36764,
A178125 ,1,1,1,2,3,1,5,11,6,1,16,45,34,10,1,61,208,197,81,15,1,269,1068,1204,626,165,21,1,1337,6017,7810,4863,1640,302,28,1,7354,36801,53762,38742,15781,3760,511,36,1,44155,242242,391797,319197,151487,43962,7805,814,
-A178126 ,1,1,2,4,6,9,9,24,56,24,16,120,250,275,50,25,720,1884,1350,960,90,36,5040,12348,14896,5145,2695,147,49,40320,114624,105056,80416,15680,6496,224,64,362880,986256,1282284,605556,336609,40824,13986,324,81,3628800,
+A178126 ,1,2,4,6,9,9,24,56,24,16,120,250,275,50,25,720,1884,1350,960,90,36,5040,12348,14896,5145,2695,147,49,40320,114624,105056,80416,15680,6496,224,64,362880,986256,1282284,605556,336609,40824,13986,324,81,
A178127 ,149,179,227,239,347,431,569,599,641,821,1019,1049,1061,1427,1487,1607,1787,1997,2081,2129,2237,2267,2657,2687,2711,2789,2999,3167,3257,3299,3359,3527,3539,3581,3671,3917,4091,4127,4229,4241,4337,4547,4637,4649,
A178128 ,11,17,29,41,59,71,101,107,149,179,227,239,269,281,311,347,419,431,461,569,599,641,659,809,821,827,857,881,1019,1031,1049,1061,1091,1151,1229,1277,1289,1301,1427,1451,1481,1487,1607,1667,1721,1787,1871,1877,1997,
A178129 ,0,2,8,23,47,87,147,224,328,463,623,821,1049,1322,1644,2004,2420,2896,3418,4007,4647,5361,6153,7004,7940,8940,10032,11220,12480,13843,15313,16863,18527,20276,22146,24141,26229,28449,30767,33224,35824,38530,
@@ -178990,7 +178990,7 @@ A178985 ,3,19,11,227,1019,269201,186023729,457933343698297657,226760286222021349
A178986 ,1,0,0,0,0,0,0,0,0,0,44,0,1092,0,0,0,16932,0,24776,0,0,0,1881492,0,
A178987 ,0,-1,-2,0,16,80,288,896,2560,6912,17920,45056,110592,266240,630784,1474560,3407872,7798784,17694720,39845888,89128960,198180864,438304768,964689920,2113929216,4613734400,10032775168,21743271936,
A178988 ,7,5,7,5,5,2,2,1,2,8,1,0,1,1,4,9,2,9,7,6,9,2,0,8,0,5,6,3,0,6,4,4,5,8,0,9,2,7,0,3,7,5,3,2,6,1,9,3,9,2,9,2,1,4,7,5,9,1,2,9,9,2,1,3,9,5,2,4,5,6,5,1,0,6,0,2,5,9,4,9,6,8,8,5,3,3,6,9,9,2,8,4,4,4,9,8,4,2,5,7,
-A178989 ,1,1,14,1128,90942080,57157560576,67818988957718528,115047995548743401472,674758653138775267142795264,40819609745761407890621234130376982528,
+A178989 ,1,1,14,1128,90942080,57157560576,67818988957718528,115047995548743401472,674758653138775267142795264,40819609745761407890621234130376982528,221388314080552960064314183934017536000000,79870389582370042643423622863118514819531536385179648,
A178990 ,0,1,1,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1924,1924,0,0,0,0,19799,68302,19799,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3219407612,25797991623,25797991623,3219407612,0,0,0,0,0,0,
A178991 ,2,3,5,7,11,17,43,59,67,313,443,449,619,991,1051,1601,2143,2593,2609,2753,3169,6829,20749,24917,28661,38447,49393,54323,56873,75029,372121,974177,1346273,1346333,1718369,1806209,2178313,2178373,3524603,3525019,6683821,
A178992 ,0,1,2,3,5,6,10,11,13,21,22,26,27,43,45,53,54,86,90,91,107,109,173,181,182,214,218,346,347,363,365,429,437,693,694,726,730,858,859,875,1387,1389,1453,1461,1717,1718,1750,2774,2778,2906,2907,2923,3435,3437,3501,
@@ -179432,7 +179432,7 @@ A179427 ,0,0,0,0,0,3420,576856,19760512,270487188,2209065700,12914201256,5965985
A179428 ,0,0,0,0,0,486,346381,36285336,956078397,12428297150,104000525596,643409498286,3191250652226,13361641961066,48905750870775,160414160371552,480243686391743,1330654487994234,3449609146025210,8439769551278350,19624142987739108,43616849672119790,93112709811981557,191696927842663704,381920049400830625,738532765420347014,1389708580432837752,2550402748009811870,4573836436177381798,8029626473495462850,
A179429 ,3,5,7,11,13,17,19,23,29,31,37,41,43,47,59,61,71,73,79,89,97,103,107,113,131,149,151,157,167,173,179,181,191,199,233,239,241,251,257,269,293,379,383,401,419,433,467,479,487,521,523,613,617,619,
A179430 ,1,1,1,3,9,1,84,405,81,1,17550,121500,32805,729,1,25621596,247203171,82255257,2539107,6561,1,268715232324,3543210805275,1382411964132,53628242751,199290375,59049,1,21091830512086620,373203783345533355,
-A179431 ,1,1,3,84,17550,25621596,268715232324,21091830512086620,12814543323816738705045,61742372998425082372103866380,2399699340005498870742886195375900380,
+A179431 ,1,1,3,84,17550,25621596,268715232324,21091830512086620,12814543323816738705045,61742372998425082372103866380,2399699340005498870742886195375900380,761689137813999393167583510790986701377432464,1992997938492157367948224731863936229108552184201415196,
A179432 ,1,2,15,816,316251,873642672,17743125256857,2739097835911193328,3301626910467952067341626,31698997711344336177849363574320,2460103385023594223069956382123378560008,
A179433 ,1,9,405,121500,247203171,3543210805275,373203783345533355,299059356226224581923626,1870707073035678423776605220985,93075349691648156957700437094276630105,
A179434 ,1,2,13,571,172585,357625693,5248165593907,566958191345077996,465798195439736703244606,2982999334066325867630228374270,151658307264909973462110073089257457502,
@@ -179442,7 +179442,7 @@ A179437 ,0,1,1,1,1,1,3,3,5,9,9,13,15,15,17,21,25,25,29,31,31,35,37,41,47,49,49,5
A179438 ,1,1,1,2,1,1,4,2,1,1,1,9,4,2,2,1,1,1,20,9,4,3,
A179439 ,2,3,4,5,6,8,9,10,11,12,13,14,15,17,20,21,22,24,25,26,27,29,30,33,34,35,36,37,38,39,40,41,42,43,44,45,46,48,49,50,51,52,53,54,55,56,57,58,59,61,62,64,65,66,67,68,69,70,71,72,73,75,76,77,78,79,80,81,82,83,84,85,
A179440 ,240,395,450,733,
-A179441 ,1,21,121,432,1182,2723,5558,10368,18039,29689,46695,70720,103740,148071,206396,281792,377757,498237,647653,830928,1053514,1321419,1641234,
+A179441 ,1,21,121,432,1182,2723,5558,10368,18039,29689,46695,70720,103740,148071,206396,281792,377757,498237,647653,830928,1053514,1321419,1641234,2020160,2466035,2987361,3593331,4293856,5099592,6021967,7073208,8266368,9615353,11134949,12840849,
A179442 ,2,3,16,180,3456,100800,4147200,228614400,16257024000,1448500838400,158018273280000,20713561989120000,3212195459235840000,581636820654489600000,121600871304831959040000,
A179443 ,8,9,12,14,16,20,22,26,30,32,34,35,36,38,39,44,45,48,50,51,55,56,58,62,65,68,72,74,77,78,80,84,85,86,87,90,92,93,95,96,98,100,104,108,110,111,112,114,116,119,121,122,124,125,126,128,133,134,135,140,141,
A179444 ,81,91,121,141,161,201,221,261,301,321,341,351,361,381,391,441,451,481,501,511,551,561,581,621,651,681,721,741,771,781,801,841,851,861,871,901,921,931,951,961,981,1001,1041,1081,1101,1111,1121,1141,1161,1191,1211,1221,1241,1251,1261,1281,1331,
@@ -179818,12 +179818,12 @@ A179813 ,2,3,5,6,7,8,10,11,17,18,23,24,27,35,45,55,56,76,78,84,111,114,115,117,1
A179814 ,181,3787,174692,685700,2178889,5931641,31622776,64631634,1691869691,2597429617,16328969210,22469029417,54353589638,380636413501,2506650894908,11290681881873,12924394402851,127673846293724,
A179815 ,1,3,21,163,1259,9657,73949,566797,4352755,33501979,258431853,1997743677,15473296249,
A179816 ,17,60,52,68,131,112,128,223,172,97,420,113,127,407,149,308,330,352,181,780,0,211,679,472,241,508,532,548,564,293,307,941,0,668,696,712,367,752,772,397,810,419,421,1303,892,457,1391,479,487,990,1012,0,1044,0,
-A179817 ,2,4,8,14,27,48,86,151,269,460,808,1386,2372,4048,6890,11661,19719,33167,
+A179817 ,1,2,4,8,14,27,48,86,151,269,460,808,1386,2372,4048,6890,11661,19719,33167,55705,93288,155954,260040,432895,719252,1192989,1975724,3267513,5396171,8900534,14663096,
A179818 ,17,131,223,97,113,127,149,181,211,241,293,307,941,367,397,419,421,1303,457,479,487,557,587,631,1931,661,683,691,719,727,743,773,787,797,809,811,839,863,877,907,929,937,953,967,983,1009,1021,1049,1051,1087,1117,
A179819 ,10,20,25,35,45,50,60,70,75,85,95,100,110,120,125,135,145,150,160,170,175,185,195,200,210,220,225,235,245,250,260,270,275,285,295,300,310,320,325,335,345,350,360,370,375,385,395,400,410,420,425,435,445,450,460,470,475,485,495,500,510,520,525,535,545,550,560,
A179820 ,0,1,3,1,4,1,5,1,6,1,7,1,8,1,9,1,10,1,11,1,12,1,13,1,14,1,15,1,16,1,17,1,18,1,19,1,20,1,21,1,22,1,23,1,24,1,25,1,26,1,27,1,28,1,29,1,30,1,31,1,32,1,33,1,34,1,35,1,36,1,37,1,38,1,39,1,40,1,41,1,42,1,43,1,44,1,45,
A179821 ,0,1,2,2,4,5,4,3,8,9,10,10,8,9,6,4,16,17,18,18,20,21,20,11,16,17,18,18,12,13,8,5,32,33,34,34,36,37,36,19,40,41,42,42,40,41,22,20,32,33,34,34,36,37,36,19,24,25,26,26,16,17,10,6,64,65,66,66,68,69,68,35,72,73,74,74,
-A179822 ,1,1,2,3,5,7,12,16,26,37,58,79,128,171,271,376,576,783,1239,
+A179822 ,1,1,2,3,5,7,12,16,26,37,58,79,128,171,271,376,576,783,1239,1654,2567,3505,5382,7245,11247,15036,23187,31370,47672,64146,98887,131784,201340,271350,412828,551744,843285,1125417,1715207,2299452,3479341,4654468,7090529,
A179823 ,1,1,2,5,29,408,33461,38613965,3654502875938,399133058537705128729,4125636888562548868221559797461449,4657508918199804645965719872781284840798220312648198320,
A179824 ,2,24,108,320,750,1512,2744,4608,7290,11000,15972,22464,30758,41160,54000,69632,88434,110808,137180,168000,203742,244904,292008,345600,406250,474552,551124,636608,731670,837000,953312,1081344,1221858,
A179825 ,96,402,516,786,906,1116,1146,1266,1356,3246,4206,
@@ -179839,7 +179839,7 @@ A179834 ,1,2,3,4,6,7,10,11,12,14,17,18,19,22,23,24,26,27,28,30,31,32,34,35,36,37
A179835 ,0,0,1,0,2,2,2,0,0,3,3,6,3,0,0,1,7,4,8,3,0,0,7,4,6,6,2,2,0,0,6,8,7,12,0,2,1,0,6,1,2,4,8,15,15,0,1,0,0,0,6,0,0,2,1,1,0,0,6,0,0,0,19,7,0,0,17,5,29,3,15,15,5,1,5,20,20,4,7,2,21,2,21,4,3,5,4,27,3,0,0,5,28,2,0,0,0,21,0,30,
A179836 ,4,1,8,1,5,5,4,4,9,1,4,1,3,2,1,6,7,6,6,8,9,2,7,4,2,3,9,8,4,3,3,6,1,0,6,0,8,3,5,9,5,0,1,8,6,9,0,1,0,3,8,6,2,0,8,1,7,1,9,8,3,5,0,1,7,7,6,0,4,8,5,4,
A179837 ,1,1,1,1,7,1,1,26,13,1,1,70,87,19,1,1,155,403,184,25,1,1,301,1462,1216,317,31,1,1,532,4446,6190,2725,486,37,1,1,876,11826,25954,17903,5146,691,43,1,1,1365,28314,93536,96055,41461,8695,932,49,1,1,2035,
-A179838 ,1,1,1,1,18,1,1,129,38,1,1,571,627,58,1,1,1884,6212,1525,78,1,1,5103,43123,24576,2823,98,1,1,11998,230241,277500,63660,4521,118,1,1,25362,
+A179838 ,1,1,1,1,18,1,1,129,38,1,1,571,627,58,1,1,1884,6212,1525,78,1,1,5103,43123,24576,2823,98,1,1,11998,230241,277500,63660,4521,118,1,1,25362,1005267,2379096,1014681,131464,6619,138,1,1,49347,3744753,16359996,12301986,2724266,235988,9117,158,1,
A179839 ,341,731,1333,1387,1727,2047,2701,3277,3503,3763,4033,4369,4681,5461,7957,8321,9509,10261,10669,13747,14491,15709,17557,17861,18721,19147,19951,20737,23377,31417,31609,31621,35333,42799,43921,44669,46979,49141,49901,49981,
A179840 ,1,2,3,3,4,5,4,6,6,6,7,8,8,7,7,7,10,11,10,8,8,10,12,13,12,11,11,13,14,14,11,11,11,10,11,14,15,14,13,15,17,17,13,10,15,20,21,18,14,13,15,19,19,18,16,21,21,18,16,18,17,19,21,23,25,23,17,18,20,20,21,20,23,25,28,
A179841 ,3,4,4,5,5,5,6,6,7,6,7,8,9,8,9,10,11,10,10,10,11,12,10,10,10,12,11,13,14,13,14,12,14,14,15,17,17,17,18,20,20,18,19,18,18,17,19,18,19,20,20,19,20,17,19,18,19,21,21,19,20,18,23,22,21,20,24,26,26,27,23,22,28,29,
@@ -179945,7 +179945,7 @@ A179940 ,1,2,2,3,2,4,2,4,3,4,0,4,0,2,2,3,0,4,0,4,2,0,0,4,1,0,2,2,0,4,0,2,0,0,2,3
A179941 ,1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9,
A179942 ,1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9,
A179943 ,1,1,2,1,3,3,1,4,8,4,1,5,15,21,5,1,6,24,56,55,6,1,7,35,115,209,144,7,1,8,48,204,551,780,377,8,1,9,63,329,1189,2640,2911,987,9,1,10,80,496,2255,6930,12649,10864,2584,10,1,11,99,711,3905,15456,40391,60605,40545,6765,11,
-A179944 ,1,3,7,17,47,148,518,1977,8138,35879,168500,838944,
+A179944 ,1,3,7,17,47,148,518,1977,8138,35879,168500,838944,4409957,24385913,141412615,857611641,5426144191,35739397738,244573978098,1735854397529,12757309001222,96941738970957,760649367654460,6155205917196408,51308394497243469,440110582561558831,
A179945 ,30,42,60,78,102,138,186,198,216,222,228,240,246,258,270,282,360,372,390,414,438,492,498,546,582,600,606,642,708,720,756,762,774,786,810,852,870,930,942,954,1002,1014,1020,1026,1038,1068,1086,1182,1266,1290,
A179946 ,0,1,2,1,2,8,1,2,2,108,1,2,1,1,2,1,6,2,8,1,1,9,1,1,2,17,1,5,2,2,1,1,18,6,1,2,4,3,3,3,17,2,1,2,2,1,1,4,1,22,14,1,1,1,1,2,6,1,13,1,4,2,2,3,1,13,1,2,1,5,3,2,1,6,1,3,11,5,1,9,2,1,1,4,4,2,2,1,1,1,8,1,1,4,5,111,21,2,3,
A179947 ,0,5,1,1,1,1,1,2,1,7,1,1,36,2,1,1,56,2,44,1,1,1,34,3,5,1,1,1,15,3,1,2,12,2,4,2,17,4,2,1,2,2,2,1,1,1,29,6,1,4,1,3,1,1,4,3,2,1,3,1076,17,3,49,1,2,2,2,3,20,4,13,3,2,90,1,1,2,12,2,1,3,1,7,1,10,1,2,1,2,4,206,2,2,1,6,
@@ -180167,7 +180167,7 @@ A180162 ,1,2,3,7,510,21,17490,93,217,381,651,118879530,2667,8191,11811,24573,573
A180163 ,62480,1432640,7660880,27931280,39685376,116636864,179299575,318523136,4217802560,4494828240,4952759175,6067699000,7775676090,12285798525,15069863936,17358731325,20160203840,25845386480,30293400832,
A180164 ,504,2394,5544,10584,12600,21600,26880,35712,139104,133920,138240,157248,168480,224640,262080,245520,294840,311040,348192,357120,388800,399168,645624,698544,749952,756000,892800,955206,1017792,1048320,
A180165 ,1,1,2,1,3,3,1,4,8,5,1,5,15,22,8,1,6,24,57,60,13,1,7,35,116,216,164,21,1,8,48,205,560,819,448,34,1,9,63,330,1200,2704,3105,1224,55,1,10,80,497,2268,7025,13056,11772,3344,89,1,11,99,712,3920,15588,41125,63040,44631,9136,144,
-A180166 ,1,3,7,18,51,161,560,2163,8691,38142,178107,885041,4636948,
+A180166 ,1,3,7,18,51,161,560,2123,8691,38142,178407,885041,4636948,25564727,147848651,894448186,5646589363,37115577265,253517232120,1796241061843,13180234725987,100009217354694,783656713398383,6333420109604593,52732283687195340,451831859926030943,
A180167 ,1,7,48,330,2268,15588,107136,736344,5060880,34783344,239065344,1643092128,11292944832,77616221760,533454999552,3666427327872,25199293964544,173194327754496,1190361730314240,8181336348412416,56230188472359936,386469148924634112,
A180168 ,1,3,11,37,129,443,1531,5277,18209,62803,216651,747317,2577889,8892363,30674171,105810157,364991169,1259033123,4343022091,14981209797,51677530049,178261109083,614909868411,2121125282237,7316799906529,25239226224243,87062451981131,
A180169 ,1,3,4,8,1296,32,46656,128,256,512,1024,362797056,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,
@@ -180415,14 +180415,14 @@ A180410 ,1,2,3,4,5,6,7,8,9,1,1,12,13,14,15,16,17,18,19,2,12,2,23,24,25,26,27,28,
A180411 ,16,21,24,30,32,31,37,42,41,48,39,48,45,56,45,54,51,51,61,72,59,57,55,80,71,64,65,78,61,96,70,77,75,69,91,90,71,67,87,80,101,120,87,75,128,77,101,93,72,114,121,87,81,91,152,81,126,111,113,107,90,78,168,103,93,129,123,176,
A180412 ,1,2,3,4,5,6,7,8,9,20,31,40,51,60,71,80,91,200,311,400,511,600,711,800,911,2000,3111,
A180413 ,0,144,576,1440,2880,5040,8064,12096,17280,23760,31680,41184,52416,65520,80640,97920,117504,139536,164160,191520,221760,255024,291456,331200,374400,421200,471744,526176,584640,647280,714240,785664,861696,
-A180414 ,1,2,4,8,16,36,80,194,506,1400,4039,12044,36406,111324,342447,1064835,3341434,10583931,
+A180414 ,1,2,4,8,16,36,80,194,506,1400,4039,12044,36406,111324,342447,1064835,3341434,10583931,33728050,
A180415 ,1,3,6,11,19,31,48,71,101,139,186,243,311,391,484,591,713,851,1006,1179,1371,1583,1816,2071,2349,2651,2978,3331,3711,4119,4556,5023,5521,6051,6614,7211,7843,8511,9216,9959,10741,11563,12426,13331,14279,15271,16308,
A180416 ,3,33,298,2649,23711,215341,1982296,18447847,173197435,1637524156,15570196516,148735628858,1426303768587,13722207893214,132387231596281,1280309591127436,
A180417 ,0,0,1,173,1211969509,5547480986860602794895774677,708720364531529518355420122993246286974247836241724513772950684967495246261,
A180418 ,1,3,39,32163,1720635,12345020175,1530993953307,44148864630732711,797213247855503373843915,281095572810489332134542303,26242778669866462496740532647355475,
A180419 ,1,10,5052,14240070,3152221563324450,157195096511273995860,2374214683408467590063771983920,618146855974818638210995488847340730,144946467754033586465978879886385830380958862710,
A180420 ,1,2,12,160,4592,276496,34174592,8570174016,4335215019520,4408454839564672,8992935435667848448,36753720073439398166016,300717909357395506394597376,4923649248081508021291300507648,
-A180421 ,113,131,151,199,311,337,353,359,373,733,757,919,953,991,1031,1103,1213,1217,1231,1237,1259,1301,1321,1381,1439,1471,1499,1619,1723,1741,1831,1949,3011,3019,3109,3121,3163,3257,3271,3299,3347,3527,3583,3613,3767,
+A180421 ,11,101,113,131,151,199,311,337,353,359,373,733,757,919,953,991,1031,1103,1213,1217,1231,1237,1259,1301,1321,1381,1439,1471,1499,1619,1723,1741,1831,1949,3011,3019,3109,3121,3163,3257,3271,3299,3347,3527,3583,3613,3767,
A180422 ,2,3,7,11,19,31,53,83,139,229,373,607,983,1583,2579,4177,6763,10939,17707,28649,46351,75017,121379,196387,317797,514219,832003,1346249,2178283,3524569,5702867,9227443,14930341,24157811,39088157,63245971,
A180423 ,2,28,9906,43803136,
A180424 ,1,1,0,1,1,0,1,1,1,0,1,2,2,1,0,1,2,1,2,1,0,1,3,3,3,3,1,0,1,3,3,4,3,3,1,0,1,4,3,6,6,3,4,1,0,1,4,4,8,5,8,4,4,1,0,1,5,5,10,10,10,10,5,5,1,0,1,5,4,12,10,17,10,12,4,5,1,0,
@@ -180474,7 +180474,7 @@ A180469 ,701,1301,1901,3701,6101,6701,7901,10301,13901,15101,16301,19301,21101,2
A180470 ,2,7,13,23,41,53,71,83,107,137,149,189,209,225,245,293,323,339,375,395,417,467,493,527,575,607,629,653,677,709,801,835,875,891,947,965,1023,1065,1109,1129,1193,1227,1289,1295,1333,1353,1415,1517,1555,1571,1627,
A180471 ,31,257,73,89,683,113,11,151,331,73,109,61681,127,337,5419,178481,2796203,157,1613,233,1103,2089,3033169,1321,20857,599479,281,86171,122921,19,37,109,433,38737,2731,8191,121369,22366891,13367,164511353,8831418697,23,353,397,683,2113,2931542417,
A180472 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,2,2,2,0,0,0,0,0,0,3,4,4,3,0,0,0,0,0,0,4,6,10,6,4,0,0,0,0,0,0,5,10,16,16,10,5,0,0,0,0,0,0,7,14,28,30,28,14,7,0,0,0,0,0,0,8,20,42,56,56,42,20,8,0,0,0,0,0,0,10,26,64,91,113,91,64,26,10,0,0,0,0,0,0,12,35,90,150,197,197,150,90,35,12,0,0,0,0,0,0,14,44,126,224,340,370,340,224,126,44,14,0,0,0,0,0,0,16,56,168,336,544,680,680,544,336,168,56,16,0,0,0,
-A180473 ,1,2,7,27,114,509,2365,11318,55411,276231,1397430,7156089,37023225,193229466,1016141199,5378940051,28638955098,153267403397,824014568581,4448456379134,
+A180473 ,1,2,7,27,114,509,2365,11318,55411,276231,1397430,7156089,37023225,193229466,1016141199,5378940051,28638955098,153267403397,824014568581,4448456379134,24104579252971,131055735586767,714741620026542,3908997981612017,21434123083817329,
A180474 ,2,3,5,17,23,41,47,113,131,137,149,251,263,281,293,311,317,449,503,659,677,827,881,887,1409,1787,1889,1913,2003,2069,2081,2267,2393,2531,2591,2657,2729,3083,3221,3329,3347,3767,4001,4211,4229,4583,4931,4967,5333,
A180475 ,41,271,3251,1424771,6448511,115925123,229448831,18425794691,38581737743,48264295811,73443083699,996266439503,1258302388991,1752012093443,2159450038451,2909420102783,3201110256371,18248780996099,
A180476 ,0,0,1,10,518,1,154,120,1,2,8,15,911,226,24,9470,189,2766,8224,4998,1730,49,106,3114,2030,155,231,4,119,195,2354,31,1749,29,7,2806,11704,11,1380,561,140,553,431,50231,65,7,1003,1,1905,57,456,77,231,3346,35,301,99,106,20,1045,71,280,1169,231,685,440,566,385,7994,4095,
@@ -180516,7 +180516,7 @@ A180511 ,5,51,585,95325,1290555,252645135,3616814565,764877654105,24847446219975
A180512 ,1,2,6,1,24,16,2,120,200,94,14,1,720,2400,2684,1284,310,36,2,5040,29400,63308,66158,38390,13037,2660,328,26,1,
A180513 ,111111111,111111112,111111113,111111114,111111115,111111116,111111117,111111118,111111119,111111211,111111222,111111223,111111224,111111225,111111226,111111227,111111228,111111229,111111311,111111322,111111333,111111334,111111335,111111336,111111337,111111338,111111339,111111411,111111422,111111433,111111444,111111445,111111446,111111447,111111448,111111449,111111511,111111522,
A180514 ,1,5,9,13,35,39,286,290,381,385,866,4376,10461,13506,19709,50925,139046,144086,188517,623114,6815124,7226204,7647853,8970817,42716373,64176516,189403472,240240118,463852538,520740373,
-A180515 ,0,0,1,0,0,3,0,0,198,0,0,15390,0,0,4611168,0,0,1829539224,0,0,1492247906784,0,0,1669958449339824,
+A180515 ,0,0,1,0,0,3,0,0,198,0,0,15390,0,0,4611168,0,0,1829539224,0,0,1492247906784,0,0,1669958449339824,0,0,2955696363525356640,0,0,7028088099915471491520,0,0,23308039026983275082311680,0,0,100343481973929775498656672000,
A180516 ,0,1,2,3,7,11,15,31,47,63,127,191,255,511,767,1023,2047,3071,4095,8191,12287,16383,32767,49151,65535,131071,196607,262143,524287,786431,1048575,2097151,3145727,4194303,8388607,12582911,16777215,33554431,50331647,67108863,134217727,201326591,268435455,536870911,805306367,1073741823,2147483647,3221225471,
A180517 ,23,107,127,211,223,227,241,271,283,401,421,503,523,809,829,1009,1013,1021,1031,1049,1091,1097,1103,1109,1123,1129,1201,1213,1229,1231,1249,1291,1297,1301,1307,1321,1327,1409,1429,1601,1607,1621,1627,2003,2011,
A180518 ,37,101,107,109,131,137,139,307,311,317,337,347,359,367,379,389,397,401,409,431,439,601,631,709,739,809,839,907,937,1019,1021,1031,1061,1201,1231,1301,1319,1321,1361,1409,1439,1607,1637,1801,1831,1901,1931,2011,
@@ -180923,7 +180923,7 @@ A180918 ,0,0,1,0,1,1,0,2,1,1,0,2,2,1,1,0,3,2,3,1,1,0,3,3,3,3,1,1,0,4,3,6,3,4,1,1
A180919 ,1,733,1467,2203,2941,3681,4423,5167,5913,6661,7411,8163,8917,9673,10431,11191,11953,12717,13483,14251,15021,15793,16567,17343,18121,18901,19683,20467,21253,22041,22831,23623,24417,25213,26011,26811,
A180920 ,1,33,2017,124993,7747521,480221281,29765971873,1845010034817,114360856186753,7088528073543841,439374379703531361,27234123013545400513,1688076252460111300417,104633493529513355225313,6485588522577367912668961,402001854906267297230250241,
A180921 ,1,2079,7876385,30254180671,116236127290689,446579144331338591,1715756954644453458529,6591937773063166150358655,25326223208345427203876398721,97303342974524967600723097592479,373839418381901692962342398114034081,
-A180922 ,8,12,102,1001,10002,100006,1000002,10000005,100000006,1000000003,10000000001,100000000006,1000000000001,10000000000001,100000000000018,1000000000000002,10000000000000006,100000000000000007,1000000000000000001,
+A180922 ,8,12,102,1001,10002,100006,1000002,10000005,100000006,1000000003,10000000001,100000000006,1000000000001,10000000000001,100000000000018,1000000000000002,10000000000000006,100000000000000007,1000000000000000001,10000000000000000007,
A180923 ,2,16,18,20,26,28,30,38,40,42,46,56,60,62,68,72,82,86,88,96,110,112,118,130,132,138,140,150,156,158,160,166,178,192,196,210,216,220,226,228,240,242,248,250,266,276,278,280,290,292,300,306,320,326,342,348,350,
A180924 ,4,24,29,47,61,63,67,69,87,101,129,143,153,249,252,333,408,561,616,732,929,1349,3467,6156,6919,9244,14413,17128,20059,20169,20512,23479,24076,26208,27189,
A180925 ,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,54,55,56,57,58,59,61,62,63,64,65,66,67,68,69,70,
@@ -180958,10 +180958,10 @@ A180953 ,1,6,15,29,48,72,100,134,172,214,262,314,371,433,500,571,647,728,813,904
A180954 ,1,32,2271,79936,2103269,49998072,1163531779,27263453288,
A180955 ,1,1,1,3,1,1,5,3,1,1,35,5,3,1,1,63,35,5,3,1,1,231,63,35,5,3,1,1,429,231,63,35,5,3,1,1,6435,429,231,63,35,5,3,1,1,12155,6435,429,231,63,35,5,3,1,1,46189,12155,6435,429,231,63,35,5,3,1,1,88179,46189,12155,6435,429,
A180956 ,1,2,1,8,2,1,16,8,2,1,128,16,8,2,1,256,128,16,8,2,1,1024,256,128,16,8,2,1,2048,1024,256,128,16,8,2,1,32768,2048,1024,256,128,16,8,2,1,65536,32768,2048,1024,256,128,16,8,2,1,262144,65536,32768,2048,1024,256,128,
-A180957 ,1,1,1,1,1,1,1,0,0,1,1,-2,-5,-2,1,1,-5,-15,-15,-5,1,1,-9,-30,-41,-30,-9,1,1,-14,-49,-77,-77,-49,-14,1,1,-20,-70,-112,-125,-112,-70,-20,1,1,-27,-90,-126,-117,-117,-126,-90,-27,1,1,-35,-105,-90,45,131,45,-90,-105,
-A180958 ,1,1,2,2,2,-1,-8,-25,-57,-114,-202,-322,-447,-496,-271,625,2914,7762,16834,32063,54760,83319,108375,103726,11110,-282498,-973439,-2366432,-4869919,-8903455,-14604094,
-A180959 ,1,1,1,1,3,1,1,6,6,1,1,10,23,10,1,1,15,65,65,15,1,1,21,150,321,150,21,1,1,28,301,1197,1197,301,28,1,1,36,546,3584,7531,3584,546,36,1,1,45,918,9114,35523,35523,9114,918,45,1,1,55,1455,20490,132045,276433,132045,
-A180960 ,1,1,1,1,4,1,1,9,9,1,1,16,46,16,1,1,25,150,150,25,1,1,36,375,952,375,36,1,1,49,791,4039,4039,791,49,1,1,64,1484,12992,31078,12992,1484,64,1,1,81,2556,34524,162774,162774,34524,2556,81,1,1,100,4125,79920,641250,
+A180957 ,1,1,1,1,1,1,1,0,0,1,1,-2,-5,-2,1,1,-5,-15,-15,-5,1,1,-9,-30,-41,-30,-9,1,1,-14,-49,-77,-77,-49,-14,1,1,-20,-70,-112,-125,-112,-70,-20,1,1,-27,-90,-126,-117,-117,-126,-90,-27,1,1,-35,-105,-90,45,131,45,-90,-105,-35,1,
+A180958 ,1,1,2,2,2,-1,-8,-25,-57,-114,-202,-322,-447,-496,-271,625,2914,7762,16834,32063,54760,83319,108375,103726,11110,-282498,-973439,-2366432,-4869919,-8903455,-14604094,-21135454,-25294718,-19009153,14697432,107405319,311830247,705982670,1386882198,2436851006,3830805953,
+A180959 ,1,1,1,1,3,1,1,6,6,1,1,10,23,10,1,1,15,65,65,15,1,1,21,150,321,150,21,1,1,28,301,1197,1197,301,28,1,1,36,546,3584,7531,3584,546,36,1,1,45,918,9114,35523,35523,9114,918,45,1,1,55,1455,20490,132045,276433,132045,20490,1455,55,1,
+A180960 ,1,1,1,1,4,1,1,9,9,1,1,16,46,16,1,1,25,150,150,25,1,1,36,375,952,375,36,1,1,49,791,4039,4039,791,49,1,1,64,1484,12992,31078,12992,1484,64,1,1,81,2556,34524,162774,162774,34524,2556,81,1,1,100,4125,79920,641250,1484504,641250,79920,4125,100,1,
A180961 ,27,37,47,57,67,71,72,73,74,75,76,78,79,87,97,101,102,103,104,105,106,108,109,110,112,201,202,203,204,205,206,208,209,210,
A180962 ,1,1,2,16,4200,1093025200,
A180963 ,21,42,69,81,84,87,93,117,138,162,168,171,174,186,213,234,261,273,276,279,285,309,321,324,327,333,336,339,342,345,348,351,357,369,372,375,381,405,426,453,465,468,471,477,501,522,546,552,555,558,570,597,618,
@@ -181090,7 +181090,7 @@ A181085 ,1,3,25,327,6336,513657,142074241,52903930911,36806786795365,14830870563
A181086 ,3,11,13,37,101,137,271,2161,4649,8779,9091,9901,27961,52579,69857,333667,459691,513239,909091,2906161,5882353,10838689,39526741,99990001,121499449,265371653,1056689261,1058313049,5363222357,5964848081,
A181087 ,1,2,1,1,3,1,2,4,1,3,1,1,1,5,2,2,1,4,1,1,2,6,2,3,1,5,1,1,3,7,2,4,1,2,2,1,6,1,1,1,1,3,3,1,1,4,8,2,5,1,2,3,1,7,1,1,1,2,3,4,1,1,5,9,2,6,1,2,4,1,8,1,1,1,3,3,5,2,2,2,1,1,6,10,1,3,3,2,7,1,1,2,2,4,4,1,2,5,1,9,1,1,1,4,3,6,2,2,3,1,1,7,11,1,3,4,2,8,1,1,
A181088 ,1,-4,-40,672,8064,-253440,-3294720,153753600,2091048960,-130025226240,-1820353167360,141707492720640,2024392753152000,-189483161695027200,-2747505844577894400,300609462994993152000,4408938790593232896000,
-A181089 ,2,2,2,2,0,2,8,-12,-12,8,28,0,-96,0,28,32,120,-160,-160,120,32,-56,0,240,0,240,0,-56,128,-1680,-1344,3360,3360,-1344,-1680,128,1936,0,-17024,0,26880,0,-17024,0,1936,512,30240,-9216,-80640,48384,48384,
+A181089 ,2,2,2,2,0,2,8,-12,-12,8,28,0,-96,0,28,32,120,-160,-160,120,32,-56,0,240,0,240,0,-56,128,-1680,-1344,3360,3360,-1344,-1680,128,1936,0,-17024,0,26880,0,-17024,0,1936,512,30240,-9216,-80640,48384,48384,-80640,-9216,30240,512,
A181090 ,1,1,9,28,126,585,2198,9632,44226,167832,704970,3543517,12649338,53609220,257397588,1000032768,4073003174,19720373400,73088555292,323884878912,1476102415284,5555586582000,23533806109394,
A181091 ,1,1,1,2,4,2,12,6,16,20,88,12,232,84,60,138,1596,144,1008,40,420,792,28656,264,3000,15080,5616,840,514228,60,335824,152214,19800,135660,141960,7632,13320,785232,135720,2160,1009256,420,433494436,94248,
A181092 ,1,2,3,5,6,7,8,9,11,12,13,14,15,16,17,20,21,22,23,24,25,26,27,28,30,31,32,33,34,35,36,37,38,39,40,42,43,44,45,46,47,48,49,50,51,52,53,54,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,72,73,74,75,76,77,78,79,80,
@@ -182295,8 +182295,8 @@ A182290 ,1,1,2,5,16,65,386,3700,55784,1134526,27053464,
A182291 ,2,4,6,2,10,12,2,16,18,2,22,4,2,28,30,2,2,36,2,40,42,2,46,6,2,52,2,2,58,60,2,6,66,2,70,72,2,2,78,2,82,6,2,88,18,2,2,96,2,100,102,2,106,108,
A182292 ,34155,407715,8415,
A182293 ,1,2,22,2418,4276864,
-A182294 ,0,0,0,0,0,1,20349,21426300,8956859646,2352103292070,470090359867986,79002015147719136,11836068369346126698,1640443794179544776604,215598057543037336382670,
-A182295 ,0,0,0,0,0,0,5985,13112470,8535294180,3096620034795,800118566011380,166591475854153740,30012638793107746776,4892304538906805158775,743352352817243899253160,
+A182294 ,0,0,0,0,0,1,20349,21426300,8956859646,2352103292070,470090359867986,79002015147719136,11836068369346126698,1640443794179544776604,215598057543037336382670,27336005392867324870778880,3385297472808136707459580488,413211903044379104303226531072,
+A182295 ,0,0,0,0,0,0,5985,13112470,8535294180,3096620034795,800118566011380,166591475854153740,30012638793107746776,4892304538906805158775,743352352817243899253160,107478174967432322995403280,15008321493306766503800761840,2046331888629918743459557040544,
A182296 ,509203,1330207,2251349,2554843,2924861,3177553,3292241,3580901,3661529,3661543,4384979,6055001,7576559,7629217,8086751,8643209,9053711,9203917,9545351,10219379,10645867,10913233,10913681,11694013,11942443,13161283,14608183,15627133,
A182297 ,21,39,55,57,105,111,147,155,165,171,183,195,201,203,205,219,231,237,253,273,285,291,301,305,309,327,333,355,357,385,399,417,429,453,465,483,489,495,497,505,507,525,543,555,579,597,605,609,615,627,633,651,655,
A182298 ,0,2,4,3,6,5,4,7,7,6,5,10,8,8,7,6,12,11,9,9,8,7,11,13,12,10,10,9,8,15,12,14,13,11,11,10,9,17,16,13,15,14,12,12,11,10,
@@ -182372,7 +182372,7 @@ A182367 ,1,3,67,21350,147512732,30761087800216,
A182368 ,1,0,1,-4,6,-3,0,1,-12,66,-216,459,-648,594,-323,79,0,1,-24,276,-2015,10437,-40614,122662,-292883,557782,-848056,1022204,-960627,682349,-346274,112275,-17493,0,1,-40,780,-9864,90798,-647352,3714180,-17590911,69997383,
A182369 ,8,6,7,5,3,0,9,0,1,9,8,1,6,8,5,4,0,9,7,5,5,8,2,7,5,2,2,4,9,6,1,4,3,1,8,3,8,4,4,0,2,9,7,2,3,1,3,2,8,1,1,6,9,3,7,7,1,5,6,5,8,9,5,6,1,7,6,0,6,0,3,9,0,3,5,9,1,8,9,7,8,3,5,4,0,3,1,2,6,0,6,4,5,9,5,0,5,4,2,7,9,7,1,3,6,8,9,8,
A182370 ,1,1,20,3246670537110000,
-A182371 ,0,0,0,0,0,0,1330,6905220,7279892361,3717889913655,1255470137209650,326123611416074340,70993993399632155710,13659118629343706026053,2405832308811599670396135,397496768417871214784702640,
+A182371 ,0,0,0,0,0,0,1330,6905220,7279892361,3717889913655,1255470137209650,326123611416074340,70993993399632155710,13659118629343706026053,2405832308811599670396135,397496768417871214784702640,62693059156926401902640364120,9561367292987041683030275944320,
A182372 ,2,2,3,4,6,8,11,14,19,24,31,39,50,61,77,94,117,141,173,208,253,302,363,431,516,609,723,850,1003,1174,1379,1607,1878,2181,2537,2936,3404,3925,4532,5212,5998,6877,7890,9021,10320,11771,13427,15277,17385,19734,22401,25375,28739,32485,
A182373 ,3,5,7,37,45,53,179,277,721,2087,6197,6317,8775,12781,
A182374 ,19,52489,59296646043258913,3140085798164163223281069127,281013956365219695455558985684629594690518822413326510467,
@@ -185229,8 +185229,8 @@ A185224 ,0,0,0,0,0,0,0,0,1,1,2,2,4,4,6,7,10,11,15,17,23,26,33,38,49,56,69,80,99,
A185225 ,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,5,6,8,9,12,14,17,20,25,29,35,41,49,57,69,79,94,109,128,149,175,201,235,271,316,363,422,483,559,642,739,846,974,1111,1276,1455,1665,1896,2167,2463,2808,3188,3626,4111,4672,5286,5994,6777,7670,8661,9790,11036,
A185226 ,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,5,5,7,8,10,11,15,16,20,23,28,31,39,43,52,59,70,79,95,106,125,142,166,187,220,247,287,325,375,423,490,551,633,715,818,921,1055,1186,1352,1522,1729,1943,2208,
A185227 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31,35,41,47,55,62,72,82,95,107,124,140,161,182,208,235,269,303,345,389,442,497,564,634,718,806,910,1021,1152,1290,1452,1627,1828,2044,2294,
-A185228 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,6,6,8,9,11,12,15,16,20,22,26,29,35,38,45,50,59,65,76,84,98,109,125,139,161,178,204,227,259,288,328,364,414,460,520,578,654,725,
-A185229 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,5,6,7,8,10,11,13,15,17,19,23,25,29,33,38,42,49,54,62,70,79,88,101,112,
+A185228 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,6,6,8,9,11,12,15,16,20,22,26,29,35,38,45,50,59,65,76,84,98,109,125,139,161,178,204,227,259,288,328,364,414,460,520,578,654,725,817,908,1021,1133,
+A185229 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,5,6,7,8,10,11,13,15,17,19,23,25,29,33,38,42,49,54,62,70,79,88,101,112,127,142,160,178,202,224,252,281,315,350,394,436,488,543,606,672,751,831,926,1027,1142,
A185230 ,5,67,157,12211,313553,
A185231 ,1,2,3,5,7,13,23,43,83,163,317,631,1259,2503,5003,9973,19937,39869,79699,159389,318751,637499,1274989,2549951,5099893,10199767,20399531,40799041,81598067,163196129,326392249,652784471,1305568919,2611137817,
A185232 ,0,0,0,0,4,0,0,0,1520,0,0,0,235072,0,0,705280,278539264,0,0,0,226593936,0,0,0,295266178368,0,24143851798528,27,10557680820452065280,0,0,0,2821525007683005301391360,0,0,2821525007683005301391360,43942858408664114852524638339072,
@@ -186187,7 +186187,7 @@ A186182 ,1,1,2,8,50,388,3363,31132,301156,3007000,30753169,320492869,3391067666,
A186183 ,1,1,2,9,68,646,6857,77695,919642,11233858,140544189,1791614714,23187320736,303861373679,4023883823059,53762917329659,723854999871943,9811154512175468,133762940465746744,1833187046654598058,25239961633188882896,
A186184 ,1,1,2,10,89,1002,12592,168805,2363241,34138860,505042286,7612594936,116492572621,1804984878387,28260999959595,446441276449715,7106718529937710,113886198966545724,
A186185 ,1,1,3,11,48,239,1306,7612,46436,292875,1894365,12496864,83753165,568628232,3902600850,27031069848,188709211952,1326456525471,9379857716098,66680723764051,476269444919163,3416178576731504,
-A186186 ,1,1,3,12,63,403,2919,22833,187799,1599718,13984383,124717327,1130144932,10375309228,96290993853,901915801437,8514822062757,80939662475426,
+A186186 ,1,1,3,12,63,403,2919,22833,187799,1599718,13984383,124717327,1130144932,10375309228,96290993853,901915801437,8514822062757,80939662475426,774025387921462,7441380898249458,71879194326339456,697253570563306939,6789448668631285664,66340474776507262638,
A186187 ,1,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,2,1,2,4,2,1,2,2,
A186188 ,1,1,1,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
A186189 ,1,1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,
@@ -188227,7 +188227,7 @@ A188222 ,1,2,3,5,6,7,9,10,11,13,14,15,17,18,19,20,22,23,24,26,27,28,30,31,32,34,
A188223 ,1,1,3,5,9,16,29,53,96,174,315,571,1035,1876,3400,6162,11168,20241,36685,66488,120503,218400,395829,717402,1300222,2356527,4270978,7740736,14029338,25426823,46083666,83522203,151375943,274354307,497240739,901200915,1633339800,2960270965,5365205811,
A188224 ,1,2,6,10,24,54,96,200,442,650,1548,2370,4060,7670,15792,25740,49074,81900,147756,251012,
A188225 ,2,31,256,1496,6936,27132,93024,286824,810084,2124694,5230016,12183560,27041560,57500460,117675360,232676280,445962870,830905245,1508593920,2674776720,4639918800,7887861960,13160496960,21578373360,34810394760,
-A188226 ,1,7,63,315,945,1575,3465,19845,10395,17325,26775,127575,45045,266805,190575,155925,135135,2480625,225225,130203045,405405,1289925,2168775,1715175,675675,3898125,3468465,1576575,3239775,67798585575,2027025,16769025,2297295,20539575,42170625,27286875,
+A188226 ,1,7,63,315,945,1575,3465,19845,10395,17325,26775,127575,45045,266805,190575,155925,135135,2480625,225225,130203045,405405,1289925,2168775,1715175,675675,3898125,3468465,1576575,3239775,67798585575,2027025,16769025,2297295,20539575,42170625,27286875,3828825,117661005,
A188227 ,1,3,12,81,598,4859,41748,374820,3475284,33053399,320869264,3167899567,31721907390,321494911644,3292220700520,34018798556265,354312456839426,3716173628641396,39220841304052510,416264662788000213,
A188228 ,1,2,4,15,58,245,1082,5020,24040,118154,592332,3019280,15604848,81614541,431227650,2298833499,12350952158,66818754504,363727676848,1990946917066,10952512200610,60525264890259,335856363303010,1870732844007387,
A188229 ,1,3,7,30,119,527,2395,11376,55368,275735,1397063,7185356,37419881,196993233,1046785509,5608211846,30264466262,164375822152,897938065590,4930713542112,27202861579741,150721902660263,838367664692809,
@@ -192792,7 +192792,7 @@ A192787 ,0,1,3,3,2,8,7,10,6,12,9,21,4,17,39,28,4,26,11,36,29,25,21,57,10,20,29,4
A192788 ,3,12,12,36,48,24,24,66,132,42,114,60,48,84,216,90,168,72,108,246,42,228,162,66,48,102,156,150,96,84,198,192,108,222,114,192,144,144,402,162,306,108,408,36,150,252,186,366,216,126,126,672,54,312,120,450,180,300,198,114,204,222,252,486,108,204,228,126,228,204,
A192789 ,1,3,2,7,9,4,4,11,21,7,19,9,7,14,34,13,27,11,17,40,7,37,27,10,8,16,27,25,15,13,33,32,17,36,18,31,24,24,65,26,47,17,67,6,23,42,30,58,37,20,19,106,8,51,19,71,28,48,31,17,33,34,40,79,16,34,38,21,39,32,19,110,52,33,39,86,30,29,23,15,81,16,93,19,
A192790 ,4,80,336,880,1820,3264,5320,8096,11700,16240,21824,28560,36556,45920,56760,69184,83300,99216,117040,136880,158844,183040,209576,238560,270100,304304,341280,381136,423980,469920,
-A192791 ,40,112,240,448,760,1200,1792,2560,3528,4720,6160,7872,9880,12208,14880,17920,21352,25200,29488,34240,39480,45232,51520,58368,65800,73840,82512,91840,101848,112560,
+A192791 ,40,112,240,448,760,1200,1792,2560,3528,4720,6160,7872,9880,12208,14880,17920,21352,25200,29488,34240,39480,45232,51520,58368,65800,73840,82512,91840,101848,112560,124000,136192,149160,162928,177520,192960,209272,226480,244608,263680,
A192792 ,72,360,2556,22572,219636,2204244,22197420,222257988,2207645892,21754722852,212845625820,2069408197476,20010127994676,192565336573476,1845376043710284,17619057807964452,167667905660138532,1590879916369856484,15054743317985652924,
A192793 ,108,360,900,1872,3420,5688,8820,12960,18252,24840,32868,42480,53820,67032,82260,99648,119340,141480,166212,193680,224028,257400,293940,333792,377100,424008,474660,529200,587772,650520,
A192794 ,1,3,5,15,17,27,35,45,57,65,87,95,125,135,137,147,155,177,255,267,275,347,357,407,447,455,477,507,605,615,707,717,755,767,785,795,827,837,905,935,945,1185,1235,1247,1257,1275,1325,1365,1457,1497,1595,1695,
@@ -193073,7 +193073,7 @@ A193068 ,12,42,98,188,320,502,742,1048,1428,1890,2442,3092,3848,4718,5710,6832,8
A193069 ,24,36,71,119,120,127,143,144,145,216,240,343,354,355,360,384,456,595,660,693,713,715,719,720,721,722,723,724,725,726,727,728,729,733,736,744,799,936,
A193070 ,3,5,17,27,41,49,59,71,89,101,125,131,167,169,173,289,293,383,529,677,701,729,743,761,773,827,839,841,857,911,1091,1097,1163,1181,1193,1217,1373,1427,1487,1559,1583,1709,1811,1847,1849,1931,1973,2129,2197,2273,2309,
A193071 ,1,7,9,11,13,15,19,21,23,25,29,31,33,35,37,39,43,45,47,51,53,55,57,61,63,65,67,69,73,75,77,79,81,83,85,87,91,93,95,97,99,103,105,107,109,111,113,115,117,119,121,123,127,129,133,135,137,139,141,143,145,147,
-A193072 ,39,507,2379,6591,13167,29511,148955,1672209,8852259,212370543,
+A193072 ,39,507,2379,6591,13167,29511,148955,1672209,8852259,212370543,1929229929,
A193073 ,1,1,1,2,1,1,1,2,1,3,1,1,1,1,2,1,1,2,2,3,1,4,1,1,1,1,1,2,1,1,1,2,2,1,3,1,1,3,2,4,1,5,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,2,2,2,3,1,1,1,3,2,1,3,3,4,1,1,4,2,5,1,6,1,1,1,1,1,1,1,2,1,
A193074 ,1,1,3,11,46,208,993,4932,25246,132327,706936,3836187,21090069,117230109,657797920,3721215175,21201525478,121554438782,700793218421,4060389849633,23631280018767,138090295023370,809908779557386,4766153373556047,28134449221105339,
A193075 ,1,1,3,9,5,6,4,7,0,6,8,7,9,3,2,1,6,0,8,2,3,7,8,8,1,6,5,0,5,7,9,3,1,8,7,1,1,3,1,7,3,5,8,0,0,7,5,5,8,5,2,2,8,1,7,4,5,0,1,3,3,5,1,7,8,9,0,7,2,4,8,6,0,3,9,5,9,6,7,2,5,7,3,4,6,3,0,2,0,5,5,2,9,8,2,5,0,2,2,0,
@@ -196840,7 +196840,7 @@ A196835 ,1,4,15,51,146,273,-319,-6374,-36235,-113833,69388,3772035,28631669,1127
A196836 ,2,5,15,50,177,650,2445,9350,36177,141170,554325,2186750,8656377,34355690,136617405,544061750,2169039777,8654570210,34553579685,138020346350,551499730377,2204254480730,8811785649165,35231447872550,140878711512177,563373614503250,
A196837 ,1,2,-3,3,-12,11,4,-30,70,-50,5,-60,255,-450,274,6,-105,700,-2205,3248,-1764,7,-168,1610,-7840,20307,-26264,13068,8,-252,3276,-22680,89796,-201852,236248,-109584,9,-360,6090,-56700,316365,-1077300,2171040,-2345400,1026576,10,-495,10560,-127050,946638,-4510275,13667720,-25228500,25507152,-10628640,
A196838 ,1,-1,1,1,-1,1,0,1,-3,1,-1,0,1,-2,1,0,-1,0,5,-5,1,1,0,-1,0,5,-3,1,0,1,0,-7,0,7,-7,1,-1,0,2,0,-7,0,14,-4,1,0,-3,0,2,0,-21,0,6,-9,1,5,0,-3,0,5,0,-7,0,15,-5,1,0,5,0,-11,0,11,0,-11,0,55,-11,1,
-A196839 ,1,2,1,6,1,1,1,2,2,1,30,1,1,1,1,1,6,1,3,2,1,42,1,2,1,2,1,1,1,6,1,6,1,2,2,1,30,1,3,1,3,1,3,1,1,1,10,1,1,1,5,1,1,2,1,66,1,2,1,1,1,1,1,2,1,1,1,6,1,2,1,1,1,1,1,6,2,1,2730,1,1,1,2,1,
+A196839 ,1,2,1,6,1,1,1,2,2,1,30,1,1,1,1,1,6,1,3,2,1,42,1,2,1,2,1,1,1,6,1,6,1,2,2,1,30,1,3,1,3,1,3,1,1,1,10,1,1,1,5,1,1,2,1,66,1,2,1,1,1,1,1,2,1,1,1,6,1,2,1,1,1,1,1,6,2,1,2730,1,1,1,2,1,1,1,2,1,1,1,1,
A196840 ,1,1,1,1,1,1,0,1,1,1,-1,0,1,1,1,0,-1,0,5,1,1,1,0,-1,0,1,1,1,0,1,0,-7,0,7,1,1,-1,0,2,0,-7,0,2,1,1,0,-3,0,1,0,-7,0,3,1,1,5,0,-1,0,1,0,-1,0,5,1,1,
A196841 ,1,1,1,1,4,3,1,8,19,12,1,13,59,107,60,1,19,137,461,702,360,1,26,270,1420,3929,5274,2520,1,34,478,3580,15289,36706,44712,20160,1,43,784,7882,47509,174307,375066,422568,181440,1,53,1214,15722,126329,649397,
A196842 ,1,1,1,1,3,2,1,7,14,8,1,12,49,78,40,1,18,121,372,508,240,1,25,247,1219,3112,3796,1680,1,33,447,3195,12864,28692,32048,13440,1,42,744,7218,41619,144468,290276,301872,120960,1,52,1164,14658,113799,560658,
@@ -201144,7 +201144,7 @@ A201139 ,6,282,5252,38763,129244,120096,4264060,46991775,263910168,769159517,105
A201140 ,1,51,758,13009,120096,1268728,8360853,58395657,309819522,1599103606,6891312239,29648211631,108584697209,395584327260,1292379405268,4136956355752,12134954233672,35217094178978,
A201141 ,6,848,35810,593543,4264060,8360853,543067656,11302225941,110916509158,542120293937,1230211025824,1086877299841,
A201142 ,6,15,15,20,30,20,15,5,5,15,6,135,402,135,6,1,282,117,117,282,1,6,51,5252,7642,5252,51,6,15,848,758,38763,38763,758,848,15,20,1189,35810,13009,129244,13009,35810,1189,20,15,120,4788,593543,120096,120096,
-A201143 ,1,1,3,6,3,7,24,30,16,3,15,80,180,220,155,60,10,31,240,840,1740,2340,2106,1260,480,105,10,63,672,3360,10360,21840,33054,36757,30240,18270,7910,2331,420,35,
+A201143 ,1,1,3,6,3,7,24,30,16,3,15,80,180,220,155,60,10,31,240,840,1740,2340,2106,1260,480,105,10,63,672,3360,10360,21840,33054,36757,30240,18270,7910,2331,420,35,127,1792,12096,51520,154280,343392,586488,782944,824670,686840,450296,229656,89208,25480,5040,616,35,
A201144 ,1,2,3,5,6,7,8,9,11,13,13,13,14,19,21,21,18,19,22,29,31,31,25,25,26,33,41,43,43,36,32,33,37,46,55,57,57,49,41,41,42,51,61,71,73,73,64,55,50,51,56,67,78,89,91,91,81,71,61,61,62,73,85,97,109,111,111,100,89,78,72,73,79,92,105,118,131,133,133,121,109,97,85,85,86,99,113,127,141,155,157,157,144,131,118,105,98,99,106,121,
A201145 ,1,0,1,0,1,0,0,1,2,11,0,1,2,42,320,0,1,6,199,3278,71648,0,1,10,858,29904,1369736,55717584,0,1,22,3881,285124,27876028,2372510658,213773992667,0,1,42,17156,2671052,549405072,98927211122,18677872557034,3437213982024260,
A201146 ,1,2,1,6,3,1,6,3,1,1,30,15,5,5,1,30,15,5,5,1,1,210,105,35,35,7,7,1,210,105,35,35,7,7,1,1,210,105,35,35,7,7,1,1,1,210,105,35,35,7,7,1,1,1,1,2310,1155,385,385,77,77,11,11,11,11,1,2310,1155,385,385,
@@ -202183,7 +202183,7 @@ A202178 ,1,0,1,0,2,1,0,3,5,1,0,4,17,9,1,0,5,45,50,14,1,0,6,115,218,114,20,1,0,7,
A202179 ,1,2,1,3,5,1,4,17,9,1,5,45,50,14,1,6,115,218,114,20,1,7,278,851,709,224,27,1,8,679,3161,3818,1867,398,35,1,9,1666,11507,19042,13113,4276,657,44,1,10,4167,41837,91383,83222,37898,8845,1025,54,1,
A202180 ,1,1,3,9,31,115,474,2097,9967,50315,268442,1505463,8840306,54169431,
A202181 ,1,1,1,1,3,1,1,7,6,1,1,13,24,10,1,1,25,77,61,15,1,1,43,228,291,130,21,1,1,76,644,1229,856,246,28,1,1,128,1776,4872,4840,2136,427,36,1,1,216,4854,18711,25107,15543,4733,694,45,1,1,354,13184,70858,124167,101538,43120,9577,1071,55,1,
-A202182 ,1,2,5,15,49,180,715,3081,14217,69905,363926,1150036,69269925,
+A202182 ,1,2,5,15,49,180,715,3081,14217,69905,363926,1996922,1150036,69269925,
A202183 ,1,0,1,6,0,1,-12,24,0,1,100,-60,60,0,1,-540,960,-180,120,0,1,4158,-6300,4620,-420,210,0,1,-33600,71904,-35280,15680,-840,336,0,1,310896,-725760,557928,-136080,42840,-1512,504,0,1,-3160080,8723520,-6652800,
A202184 ,1,0,1,0,0,1,24,0,0,1,-60,120,0,0,1,240,-360,360,0,0,1,1260,1680,-1260,840,0,0,1,-12096,30240,6720,-3360,1680,0,0,1,105840,-290304,226800,20160,-7560,3024,0,0,1,-388800,2721600,-2358720,1058400,50400,-15120,
A202185 ,1,0,1,6,0,1,-24,24,0,1,170,-120,60,0,1,-1320,1380,-360,120,0,1,11816,-14280,6090,-840,210,0,1,-118944,171808,-77280,19600,-1680,336,0,1,1329156,-2249856,1181376,-292320,51660,-3024,504,0,1,-16313760,32093280,
@@ -210582,7 +210582,7 @@ A210577 ,12,16,20,21,25,26,27,30,31,34,35,36,38,40,41,42,43,45,46,48,49,50,51,55
A210578 ,6,10,12,15,16,20,21,25,26,27,28,30,31,34,35,36,38,40,41,42,43,45,46,48,49,50,51,55,56,57,60,61,62,63,64,65,66,70,71,72,73,75,76,77,78,80,81,83,84,85,86,87,88,90,91,92,93,94,97,98,99,100,101,102,
A210579 ,2,4,16,16,96,96,576,5184,31104,279936,1679616,8398080,58786560,352719360,1763596800,1763596800,14108774400,56435097600,169305292800,169305292800,169305292800,1693052928000,6772211712000,13544423424000,94810963968000,853298675712000,7679688081408000,23039064244224000,
A210580 ,2,2,1,2,1,0,2,1,2,2,1,0,1,2,1,2,0,2,1,2,2,2,1,0,1,1,2,1,2,0,0,2,1,2,0,2,2,1,2,2,1,2,1,0,1,1,0,2,1,0,1,1,2,2,1,2,0,0,1,2,1,2,0,2,0,2,1,2,0,2,2,2,1,2,2,1,1,2,1,0,1,1,0,0,2,1,0,
-A210581 ,1,2,7,23,68,200,615,1764,5060,14626,41785,117573,332475,933891,2609832,
+A210581 ,1,2,7,23,68,200,615,1764,5060,14626,41785,117573,332475,933891,2609832,7278512,
A210582 ,13,19,23,26,29,39,46,49,59,69,79,89,103,109,127,133,163,193,197,199,203,206,209,214,218,233,234,236,247,254,258,263,266,274,293,294,296,298,299,309,367,399,406,409,417,428,436,466,468,487,496,499,509,537,599,609,638,657,678,699,709,799,809,899,
A210583 ,1,4,1,3,7,1,6,6,9,4,1,1,5,4,0,6,9,5,7,3,0,8,1,8,9,5,2,2,4,7,5,7,7,6,2,9,7,8,8,8,7,2,6,2,2,9,7,1,8,7,9,7,6,1,9,4,3,8,7,2,5,0,6,6,5,3,8,5,1,7,3,8,2,8,2,8,7,9,4,0,4,9,3,8,2,6,1,5,6,7,
A210584 ,1,2,3,4,12,13,14,23,24,34,112,113,114,122,123,124,132,133,134,142,143,144,223,224,233,234,243,244,334,344,1112,1113,1114,1122,1123,1124,1132,1133,1134,1142,1143,1144,1213,1214,1222,1223,1224,1232,1233,1234,
@@ -212433,8 +212433,8 @@ A212428 ,0,18,37,57,78,100,123,147,172,198,225,253,282,312,343,375,408,442,477,5
A212429 ,1,1,2,4,48,96,1152,2304,276480,552960,6635520,13271040,33443020800,66886041600,802632499200,1605264998400,385263599616000,770527199232000,194172854206464000,388345708412928000,512616335105064960000,1025232670210129920000,
A212430 ,384,840,8676,33300,34980,37044,39984,42024,50604,53760,55056,61680,64380,71064,83520,88176,97644,103740,120204,129840,133896,148764,154524,160416,168120,173064,184800,188880,199056,207984,234744,266640,292116,307044,356184,
A212431 ,1,1,1,2,1,2,5,3,2,5,15,9,8,5,15,52,31,28,25,15,52,203,121,108,100,90,52,203,877,523,466,425,405,364,203,877,4140,2469,2202,2000,1875,1820,1624,877,4140,21147,12611,11250,10230,9525,9100,8932,7893,4140,21147,
-A212432 ,1,1,2,4,16,84,536,3912,32256,297072,
-A212433 ,1,1,2,3,13,71,470,3497,29203,271500,
+A212432 ,1,1,2,4,16,84,536,3912,32256,297072,3026112,33798720,410826624,5399704320,76317546240,1154312486400,18604815528960,318348065548800,5763746405053440,110086912964367360,2212209395234979840,46657233031296706560,1030510550216174469120,
+A212433 ,1,1,2,3,13,71,470,3497,29203,271500,2786711,31322803,382794114,5054810585,71735226535,1088920362030,17607174571553,302143065676513,5484510055766118,104999034898520903,2114467256458136473,44682676397748896010,988663144904696100347,
A212434 ,1,1,0,1,1,0,0,1,0,0,0,1,2,0,0,0,0,2,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,
A212435 ,1,-1,-3,11,57,-361,-2763,24611,250737,-2873041,-36581523,512343611,7828053417,-129570724921,-2309644635483,44110959165011,898621108880097,-19450718635716001,-445777636063460643,10784052561125704811,274613643571568682777,
A212436 ,9,3,3,0,9,2,0,7,5,5,9,8,2,0,8,5,6,3,5,4,0,4,1,0,1,7,1,4,0,8,7,4,3,5,8,9,0,2,5,8,9,4,7,9,7,9,5,0,1,3,7,6,4,4,6,2,3,8,4,3,7,8,8,4,0,7,9,0,6,7,2,1,6,6,3,3,0,1,2,4,3,4,3,0,1,7,6,7,3,6,3,0,3,2,7,4,3,3,6,3,7,4,8,7,6,
@@ -212581,8 +212581,8 @@ A212576 ,0,1,2,7,16,33,52,87,126,181,246,331,422,545,674,831,1006,1213,1428,1691
A212577 ,0,1,4,17,46,89,154,251,374,531,736,979,1268,1621,2024,2485,3026,3629,4302,5071,5914,6839,7876,8999,10216,11561,13004,14553,16246,18049,19970,22051,24254,26587,29096,31739,34524,37501,40624,43901,
A212578 ,0,1,4,13,28,55,92,147,216,309,420,561,724,923,1148,1415,1712,2057,2436,2869,3340,3871,4444,5083,5768,6525,7332,8217,9156,10179,11260,12431,13664,14993,16388,17885,19452,21127,22876,24739,26680,28741,
A212579 ,0,1,8,31,80,171,308,509,780,1137,1584,2143,2812,3615,4552,5645,6892,8321,9924,11731,13736,15967,18416,21117,24056,27269,30744,34515,38568,42943,47620,52641,57988,63701,69760,76211,83028,90259,97880,
-A212580 ,1,1,2,5,20,102,626,4458,36144,
-A212581 ,1,1,2,4,17,89,556,4011,32843,301210,
+A212580 ,1,1,2,5,20,102,626,4458,36144,328794,3316944,36755520,443828184,5800823880,81591320880,1228888215960,19733475278880,336551479543440,6075437671458000,115733952138747600,2320138519554562560,48827468196234035280,1076310620915575933440,
+A212581 ,1,1,2,4,17,89,556,4011,32843,301210,3059625,34104275,413919214,5434093341,76734218273,1159776006262,18681894258591,319512224705645,5782488507020050,110407313135273127,2218005876646727423,46767874983437110354,1032732727339665789981,
A212582 ,8,12,18,20,27,28,30,42,44,45,50,52,63,66,68,70,75,76,78,92,98,99,102,105,110,114,116,117,124,125,130,138,147,153,154,164,165,170,171,174,175,182,186,188,190,195,207,230,231,236,238,242,245,246,255,261,
A212583 ,66161,534851,3152573,
A212584 ,1,1,2,3,5,6,9,12,18,24,34,46,65,89,124,170,236,325,450,620,857,1182,1633,2253,3111,4293,5927,8180,11292,15585,21513,29693,40986,56571,78085,107778,148765,205336,283422,391200,539966,745302,1028725,1419925,1959892,
@@ -215664,7 +215664,7 @@ A215659 ,2,3,6,15,715,
A215660 ,1,5,7,10,27,38,82,108,207,278,486,644,1052,1404,2182,2880,4293,5654,8182,10692,15076,19604,27108,35000,47547,61020,81713,104236,137781,174800,228498,288360,373174,468566,601020,751036,955642,1188756,1501730,1859944,
A215661 ,1,3,14,83,554,3966,29756,230915,1838162,14926346,123157572,1029590062,8702171620,74238432924,638408311800,5528154378467,48161687414498,421848099386322,3712675503776372,32815429463428794,291169073934720940,2592569269501484836,
A215662 ,77551,89381,120811,265151,292471,301051,388231,477571,493541,778301,828601,971851,1008451,1123841,1133501,1154221,1163441,1219651,1243741,1265611,1295881,1559281,1668301,1796941,1842041,1929481,2071351,2080711,2119591,2545021,
-A215663 ,0,0,0,-3,-5,29,88,96,-79,-1828,-2319,-1476,-5774,-19201,73217,327052,-598256,-3501366,23884333,-4891825,-86432205,-127132665,1033299853,-1658989720,
+A215663 ,0,0,0,-3,-5,29,88,96,-79,-1828,-2319,-1476,-5774,-19201,73217,327052,-598255,-3501366,23884333,-4891825,-86432205,-127132665,1033299853,-1658989720,-1834784715,-17149335456,-17535487935,-174760519828,
A215664 ,3,0,6,-3,18,-15,57,-63,186,-246,621,-924,2109,-3393,7251,-12288,25146,-44115,87726,-157491,307293,-560199,1079370,-1987890,3798309,-7043040,13382817,-24927429,47191491,-88165104,166501902,-311686803,587670810,-1101562311,
A215665 ,0,-3,-3,-9,-6,-24,-9,-66,-3,-189,57,-564,360,-1749,1644,-5607,6681,-18465,25650,-62076,95415,-211878,348321,-731049,1256841,-2541468,4501572,-8881245,16046184,-31145307,57019797,-109482105,202204698,-385466112,716096199,
A215666 ,0,-3,6,-9,21,-33,72,-120,249,-432,867,-1545,3033,-5502,10644,-19539,37434,-69261,131841,-245217,464784,-867492,1639569,-3067260,5786199,-10841349,20425857,-38310246,72118920,-135356595,254667006,-478188705,899357613,
@@ -216717,18 +216717,18 @@ A216712 ,1,1,4,22,140,514,3444,23790,165932,774610,5767268,42526198,310791884,15
A216713 ,1,1,3,12,27,105,420,1242,5295,22395,72738,323268,1410684,4806675,21881721,97371786,341608239,1579726122,7123796790,25489388367,119184247992,542664427242,1969440159591,9284827569117,42584603672868,156213604844883,741154831030785,
A216714 ,0,1,3,6,14,29,60,123,249,503,1012,2032,4075,8164,16347,32719,65471,130986,262030,524137,1048376,2096887,4193953,8388143,16776600,33553616,67107783,134216296,268433559,536868399,1073738495,2147479238,4294961454,8589926853,17179858932,34359724787,68719458745,137438929639,274877875372,549755772064,
A216715 ,1,0,1,0,1,1,0,1,4,2,0,1,9,14,5,0,1,16,54,55,18,1,
-A216716 ,1,2,6,24,114,6,674,44,2,4714,294,30,2,37754,2272,276,16,2,340404,20006,2236,216,16,2,3412176,193896,20354,2200,156,16,2,37631268,2056012,206696,20738,1908,160,16,2,
+A216716 ,1,1,2,6,24,114,6,674,44,2,4714,294,30,2,37754,2272,276,16,2,340404,20006,2236,216,16,2,3412176,193896,20354,2200,156,16,2,37631268,2056012,206696,20738,1908,160,16,2,452745470,23744752,2273420,215024,21136,1616,164,16,2,
A216717 ,1,2,6,24,114,674,4714,37754,340404,3412176,37631268,
-A216718 ,1,1,5,0,1,20,3,0,1,102,14,3,0,1,627,72,17,3,0,1,4461,468,87,20,3,0,1,36155,3453,582,103,23,3,0,1,328849,28782,4395,704,120,26,3,0,1,3317272,267831,37257,5435,834,138,29,3,0,1,
+A216718 ,1,1,1,5,0,1,20,3,0,1,102,14,3,0,1,627,72,17,3,0,1,4461,468,87,20,3,0,1,36155,3453,582,103,23,3,0,1,328849,28782,4395,704,120,26,3,0,1,3317272,267831,37257,5435,834,138,29,3,0,1,
A216719 ,2,6,22,2,109,10,1,657,55,7,1,4625,356,54,4,1,37186,2723,362,44,4,1,336336,23300,2837,368,34,4,1,3379058,220997,25408,2967,330,35,4,1,37328103,2308564,249736,26964,3100,292,36,4,1,
A216720 ,2,6,22,109,657,4625,37186,336336,3379058,37328103,
A216721 ,2,10,55,356,2723,23300,220997,2308564,
A216722 ,1,0,0,1,5,0,0,0,1,18,5,0,0,0,1,95,18,6,0,0,0,1,600,84,28,7,0,0,0,1,4307,568,116,40,8,0,0,0,1,35168,4122,810,156,54,9,0,0,0,1,321609,33910,5975,1100,205,70,10,0,0,0,1,
A216723 ,0,0,5,18,84,568,4122,33910,
-A216724 ,3,3,24,0,100,15,0,5,594,108,18,0,4389,504,119,21,0,7,35744,3520,960,64,32,0,325395,31077,5238,927,207,27,0,9,3288600,288300,42050,8800,900,100,50,0,
+A216724 ,1,1,2,3,3,24,0,100,15,0,5,594,108,18,0,4389,504,119,21,0,7,35744,3520,960,64,32,0,325395,31077,5238,927,207,27,0,9,3288600,288300,42050,8800,900,100,50,0,36489992,2946141,409827,59785,9174,1518,319,33,0,11,
A216725 ,3,24,100,594,4389,35744,325395,3288600,
A216726 ,11,0,0,1,6,0,0,0,0,18,5,0,0,0,1,93,18,9,0,0,0,0,600,84,28,7,0,0,0,1,4320,512,192,0,16,0,0,0,0,35168,4122,810,156,54,9,0,0,0,1,321630,34000,5625,1400,200,0,25,0,0,0,0,
-A216727 ,1,6,18,93,600,4320,35168,321630,3257109,36199458,438126986,5736774869,80808984725,1218563192160,19587031966352,334329804180135,6039535339644630,115118210695441900,2308967760171049528,48613722701440862328,1072008447320752890459,
+A216727 ,1,6,18,93,600,4320,35168,321630,3257109,36199458,438126986,5736774869,
A216728 ,1,1,30,4410,1837080,1660289400,2778905329200,7757931431250000,33536835305077104000,212373276256391153904000,1887114765997607482496160000,22736049196010108202227823840000,361148501823912048339843750000000000,7389146090843722060953495522716592000000,190939093198813987007780146685866111584000000,
A216729 ,1,1,140,168000,812011200,11021058048000,339782903171712000,20692894514095964160000,2254632317437500000000000000,408212019690599470111653888000000,115985264066835726820369478446080000000,49409573278650211500346024173376634880000000,30404587048380414443886075636144408738201600000000,
A216730 ,22,333,32323,323232,2323232,3232323,22322232,23222322,23223223,33233233,223222322,223222323,232223222,332332332,2232223222,2232223223,2232223232,2322232223,2322322322,2332332332,3322332233,3323323323,22322232223,22322232232,22322232322,22322322232,22322322322,22323222322,23222322232,23223223223,
@@ -218036,7 +218036,7 @@ A218031 ,1,1,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-2,0,3,0,-1,0,-3,0,6,0,-4,0,-4,0,12,0,-
A218032 ,1,1,1,1,2,3,5,8,13,21,35,57,94,154,254,417,687,1129,1859,3057,5032,8277,13623,22412,36883,60684,99862,164312,270384,444899,732093,1204629,1982228,3261701,5367131,8831505,14532200,23912499,39347839,64746320,106539481,175309363,288469809,
A218033 ,1,1,1,0,-1,-2,-2,0,3,6,6,2,-6,-14,-16,-8,11,32,42,26,-18,-74,-108,-82,18,162,268,238,16,-344,-656,-664,-189,694,1570,1792,826,-1294,-3668,-4698,-2866,2110,8364,12034,8960,-2432,-18508,-30134,-26254,-910,39492,73862,73560,19120,
A218034 ,1,4,12,24,84,240,732,2184,6564,19680,59052,177144,531444,1594320,4782972,14348904,43046724,129140160,387420492,1162261464,3486784404,10460353200,31381059612,94143178824,282429536484,847288609440,2541865828332,7625597484984,22876792454964,
-A218035 ,4,2,5,3,8,5,13,9,22,16,37,27,60,43,93,65,138,94,
+A218035 ,4,2,5,3,8,5,13,9,22,16,37,27,60,43,93,65,138,94,197,131,272,177,365,233,478,300,613,379,772,471,957,577,1170,698,1413,835,1688,989,1997,1161,2342,1352,2725,1563,3148,1795,3613,2049,4122,2326,4677,2627,5280,2953,
A218036 ,4,6,9,8,12,16,10,15,20,25,12,18,24,30,36,14,21,28,35,42,49,16,24,32,40,48,56,64,18,27,36,45,54,63,72,81,20,30,40,50,60,70,80,90,100,22,33,44,55,66,77,88,99,110,121,24,36,48,60,72,84,96,108,120,
A218037 ,100,200,300,400,500,600,700,800,900,1001,1002,1003,1004,1005,1006,1007,1008,1009,1100,1200,1300,1400,1500,1600,1700,1800,1900,2001,2002,2003,2004,2005,2006,2007,2008,2009,2100,2200,2300,2400,2500,2600,2700,2800,
A218038 ,235,346,427,506,574,697,785,786,842,874,894,895,898,899,906,985,1086,1191,1211,1339,1342,1345,1406,1527,1546,1639,1735,1758,1765,1851,1866,1882,1937,1954,2118,2230,2233,2263,2298,2495,2505,2510,2554,2666,2678,2726,
@@ -221206,7 +221206,7 @@ A221201 ,0,9,10,196,720,6400,34272,242064,1460368,9610000,60194160,387459856,245
A221202 ,0,49,46,9025,58700,2518569,32085376,848789956,14068757692,311366232004,5733258186854,118412678934081,2273843945394922,45702294155340601,892403786998151272,17744284197389113081,348794591561701410272,
A221203 ,0,289,212,427716,4984812,1026177156,30374196832,3043294206016,133567666236732,10038761087929924,525060302368983108,34891927404610766400,1974298747702764055248,124298442666315631470276,
A221204 ,0,1681,976,20277009,433687328,420771471561,29037336149952,10986555692499984,
-A221205 ,0,2,5,13,35,98,280,815,2400,7131,21332,64172,193928,588273,1790235,5463018,16710426,51220671,157289397,483795067,1490241458,4596440959,14193917243,43878472986,135777758736,
+A221205 ,0,2,5,13,35,98,280,815,2400,7131,21332,64172,193928,588273,1790235,5463018,16710426,51220671,157289397,483795067,1490241458,4596440959,14193917243,43878472986,135777758736,420530985064,1303551591182,4043817556078,12553456467283,
A221206 ,2,5,12,33,93,269,788,2330,6947,20840,62834,190240,577990,1761279,5380796,16475256,50543777,155330201,478096270,1473591670,4547602623,14050145290,43453847676,134519899690,416794664987,1292425391245,4010608533780,12454122695249,
A221207 ,2,17,2593,5308417,26214401,57802753,584652423169,5566277615617,24807731101697,2128654511374337,114923510727115685920505857,626707144888223764167681638401,28901765777295687591430290881352276511750619137,
A221208 ,1,5,7,0,7,9,6,3,2,6,7,7,1,7,9,6,0,4,6,5,0,5,8,4,0,8,9,4,2,4,6,4,9,5,8,5,4,7,5,0,6,5,9,3,1,8,3,8,7,5,3,2,5,9,5,9,8,0,2,2,7,5,8,2,3,5,4,7,7,6,9,6,2,7,6,6,9,2,6,3,9,1,0,7,0,4,9,6,6,6,1,7,9,3,8,6,3,4,7,3,4,0,5,0,3,
@@ -223168,7 +223168,7 @@ A223163 ,139,3886,540154,43640576,6578580741,713054874385,107232700247240,129384
A223164 ,568,40380,22935840,6578580741,4148524583552,1726991116346936,1085230897885735392,515916486281448877215,317979410833372586944592,
A223165 ,4,10,10,40,50,40,139,500,500,139,568,3886,20000,3886,568,2134,40380,540154,540154,40380,2134,8724,359428,22935840,43640576,22935840,359428,8724,33639,3723640,767019352,6578580741,6578580741,767019352,3723640,
A223166 ,4,28,175,1244,9628,78625,664916,5762207,50849233,455055612,4118066398,37607950279,346065645808,3204942065690,29844571475285,279238344248555,2623557165610820,24739954309690413,234057667376222380,2220819602783663481,
-A223167 ,0,3,7,15,36,127,337,752,1699,3101,11585,38261,108969,314888,1052616,3214630,7956587,21949553,99877773,222744641,597394252,1932355206,7250186214,17146907276,
+A223167 ,0,3,7,15,36,127,337,752,1699,3101,11585,38261,108969,314888,1052616,3214630,7956587,21949553,99877773,222744641,597394252,1932355206,7250186214,17146907276,55160980937,155891678119,508666658004,1427745660372,
A223168 ,1,1,2,3,2,3,12,4,15,20,4,15,90,60,8,105,210,84,8,105,840,840,224,16,945,2520,1512,288,16,945,9450,12600,5040,720,32,10395,34650,27720,7920,880,32,10395,124740,207900,110880,23760,2112,64,135135,540540,540540,205920,34320,2496,64,
A223169 ,1,1,3,4,3,4,24,9,28,42,9,28,252,189,27,280,630,270,27,280,3360,3780,1080,81,3640,10920,7020,1404,81,3640,54600,81900,35100,5265,243,58240,218400,187200,56160,6480,243,58240,1048320,1965600,
A223170 ,1,1,4,5,4,5,40,16,45,72,16,45,540,432,64,585,1404,624,64,585,9360,11232,3328,256,9945,31824,21216,4352,256,9945,198900,318240,141440,21760,1024,208845,835380,742560,228480,26880,1024,208845,5012280,10024560,5940480,1370880,129024,4096,
@@ -223937,7 +223937,7 @@ A223932 ,239,10511,142177,1065625,5773556,26250443,108796955,427868778,162337460
A223933 ,3,9,9,22,54,22,46,218,218,46,86,698,1116,698,86,148,1915,4498,4498,1915,148,239,4690,15791,21334,15791,4690,239,367,10511,49646,86439,86439,49646,10511,367,541,21919,142177,316136,386495,316136,142177,21919,541,
A223934 ,2,2,2,3,2,2,7,2,17,7,5,3,3,2,109,3,101,19,229,5,2,23,23,17,107,269,2,29,2,31,37,197,107,73,37,7,59,233,3,3,7,43,43,5,2,47,269,61,43,3,53,13,3,643,13,5,151,59,2,
A223935 ,48497,48907,493747,578453,1223777,1249363,1933363,3304607,5160217,5765083,6022087,6205937,7740127,7757447,7862843,8173537,8938627,11989177,13789033,17649223,18142693,18829117,20006813,20601593,23938867,24448063,24478043,
-A223936 ,2,97,3877,4943,50741,1487159,3356117,131047091863,449627893189,906460844407,61168531626487,141835115384731,
+A223936 ,2,97,3877,4943,50741,1487159,3356117,131047091863,449627893189,906460844407,61168531626487,141835115384731,749668095960389,
A223937 ,8,4696450,7024453131396,17761740387522,155912686127038650,87598780898450312031408,2147216863131055036604400,2908950240914054780101441371333254159676520,384422969812280951687876430655304031054262132,6187047308209705064673104196645071104957480508,
A223938 ,2,3,4,5,6,13,14,17,30,40,41,51,54,73,121,137,364,446,485,638,925,1382,1478,2211,2726,5581,5678,6424,8524,10649,15990,17174,18685,18889,
A223939 ,8,187858,13080918308,26871014202,29988975981350,773478679579793136,8923646993118036400,545048444084018901462938808502760,22049455928935679528789623492181708,180819643079146957138056211903672348,
@@ -224067,7 +224067,7 @@ A224062 ,610,59792,1557606,21167501,200974242,1573171210,11060805360,72498474377
A224063 ,1163,180821,6643979,114643788,1274747540,11060805360,83942450048,591725806925,3973992584299,25579886531225,157381574935619,920939113739591,
A224064 ,4,16,16,50,160,50,130,984,984,130,296,4580,8854,4580,296,610,17723,58814,58814,17723,610,1163,59792,324702,506513,324702,59792,1163,2083,180821,1557606,3509115,3509115,1557606,180821,2083,3544,499357,6643979,
A224065 ,1,2,1,4,1,2,8,3,2,6,19,5,4,6,21,53,14,10,12,21,112,209,39,24,24,42,112,853,1253,170,72,72,84,224,853,11117,13599,1083,322,210,231,448,1706,11117,261080,288267,12516,2112,948,735,1232,3412,22234,261080,11716571,
-A224066 ,2,7,28,114,472,1988,8480,36474,157720,684404,
+A224066 ,1,2,7,28,114,472,1988,8480,36474,157720,684404,2976994,12971206,56587676,247097170,1079749976,4720841314,20649303934,90353041092,395459463960,1731251197242,7580521689750,33197447406682,145400339328566,636901149067534,2790082285204966,
A224067 ,1,5,5,13,9,5,13,5,9,5,17,9,13,13,5,9,21,9,9,17,0,5,9,5,13,13,13,13,17,5,9,9,0,9,9,13,13,5,17,17,17,17,17,17,21,9,5,5,25,13,5,0,25,13,13,17,9,5,13,9,33,9,9,0,0,21,9,33,21,9,13,5,13,9,17,
A224068 ,0,0,0,1536,122880,10813440,1348730880,261070258176,81787921367040,42364317235937280,36686317873382031360,53408511909378681470976,131046345314766385022238720,542471805171085602081503969280,3789399960645715708906355231293440,
A224069 ,1,-1,1,3,-4,1,-25,36,-12,1,543,-800,288,-32,1,-29281,43440,-16000,1920,-80,1,3781503,-5621952,2085120,-256000,11520,-192,1,-1138779265,1694113344,-629658624,77844480,-3584000,64512,-448,1,783702329343,-1166109967360,433693016064,-53730869248,2491023360,-45875200,344064,-1024,1,
@@ -224244,7 +224244,7 @@ A224239 ,1,2,3,13,77,1494,56978,4495023,669203528,187623057932,98793520541768,97
A224240 ,1,8,34,142,596,2530,10842,46766,202594,880210,
A224241 ,0,3,130456,342096,1226720,291575011,379894587,523040160,15216609776,136622606520,
A224242 ,0,4,24,44,112,480,1984,8064,32512,130560,263160,278828,340028,523264,2095104,8384512,25239472,32490836,33546240,134201344,536838144,2147418112,
-A224243 ,4,22,108,490,2164,9474,41374,180614,788676,
+A224243 ,4,22,108,490,2164,9474,41374,180614,788676,3445462,15059202,65847946,288033326,1260313930,5516051890,24147542122,105729680608,463006798298,2027839420598,8882324416302,38909820194506,170461077652718,746826223566214,3272185833672630,
A224244 ,1,1,2,2,9,17,63,261,1088,4374,24583,133861,740303,4514824,29945555,205127474,1464586617,10971233035,86410874373,708423380237,6026435657580,53117555943951,485246803230148,4589013046619689,44819208415713035,451184268041122808,
A224245 ,1,1,5,14,89,474,3499,27040,253161,2426300,27596051,323960856,4277055925,59041067344,898062119655,14172430400864,243919993681649,4347177953716080,83224487266425811,1653277176082392040,34961357216796300381,763702067489722288136,
A224246 ,1,1,3,8,41,194,1309,9022,79057,689588,7462601,80632826,1021071193,13120783948,192752054377,2848878770774,47617784530529,800500650553472,14910497765819137,281133366288649138,5803224036600349801,120681837753825004796,2734647516979262677673,62424209302423879016558,1535507329367939907583057,
@@ -228725,7 +228725,7 @@ A228720 ,1,2,2,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,8,8,9,9,9,9,10,10,10,10,11,11
A228721 ,2,1,9,9,1,1,4,8,5,7,5,1,2,8,5,5,2,6,6,9,2,3,8,5,0,3,6,8,2,9,5,6,5,2,0,1,8,9,3,8,0,1,8,5,7,9,5,6,2,5,7,4,0,7,4,6,8,2,4,6,1,2,1,4,6,1,5,4,7,1,4,8,4,4,0,0,3,4,6,2,9,9,0,3,9,6,2,4,3,7,7,7,3,9,4,8,1,9,4,7,5,8,7,5,
A228722 ,0,1,2,3,4,5,6,7,8,9,10,10,12,12,14,14,16,16,18,18,18,21,21,23,23,25,25,27,27,29,30,30,32,32,34,34,36,36,38,38,38,41,41,43,43,45,45,47,47,49,50,50,52,52,54,54,56,56,58,58,58,61,61,63,63,65,65,
A228723 ,0,1,2,3,4,5,6,7,8,9,10,12,12,14,14,16,16,18,18,21,21,21,23,23,25,25,27,27,29,29,30,32,32,34,34,36,36,38,38,41,41,41,43,43,45,45,47,47,49,49,50,52,52,54,54,56,56,58,58,61,61,61,63,63,65,65,67,
-A228724 ,0,0,0,3,7,-23,-73,-57,186,2126,3161,3885,12731,39462,-13815,-151907,1117163,5045162,-19274680,18700047,127912738,252060543,-656184524,2799754423,5292148929,27646015077,
+A228724 ,0,0,0,3,7,-23,-73,-57,186,2126,3161,3885,12731,39462,-13815,-151907,1117163,5045162,-19274680,18700047,127912738,252060543,-656184524,2799754423,5292148929,27646015077,49454963317,271968742992,
A228725 ,6,3,5,1,8,1,4,2,2,7,3,0,7,3,9,0,8,5,0,1,1,8,7,2,1,0,5,7,7,0,2,8,9,4,9,9,5,5,8,8,2,9,7,3,5,1,5,0,0,8,9,4,2,6,4,6,3,2,2,3,6,2,2,1,8,9,1,3,0,6,7,4,3,7,3,6,7,9,6,9,3,2,7,1,
A228726 ,1,1,1,2,1,1,2,1,2,9,1,1,3,1,3,9,1,5,9,28,1,1,3,1,3,10,1,5,11,28,1,5,12,28,
A228727 ,7,13,23,131,179,229,283,337,107,641,317,163,643,193,1949,523,257,2053,1021,1933,2477,773,811,401,929,6379,457,6197,5701,1747,547,1949,1291,2083,647,661,2341,709,1579,2549,2633,1721,4909,2851,857,877,5441,4441,
@@ -229756,7 +229756,7 @@ A229751 ,3,422,2347,6561,15075,32548,69198,147376,315786,680124,1468934,3174760,
A229752 ,12,1840,6809,15075,29776,57677,113330,228657,473562,1000381,2139866,4607729,9947906,21481485,46333458,99752209,214296842,459341309,982407146,2096608977,4465367730,9492005773,20140323170,42660619313,
A229753 ,50,6456,17404,32548,57677,102271,186396,354509,704530,1450667,3059672,6544921,14099310,30453719,65785684,141933029,305639306,656732227,1407958032,3011810033,6428914918,13695082991,29117602316,61795491133,
A229754 ,210,20032,41872,69198,113330,186396,314700,557578,1046550,2070144,4258696,8984046,19221426,41402836,89392212,192991026,416096878,895389192,1922585088,4119014998,8805491466,18784661532,39993300364,
-A229755 ,0,0,0,1,3,1,3,60,60,3,12,422,598,422,12,50,1840,2347,2347,1840,50,210,6456,6809,6561,6809,6456,210,861,20032,17404,15075,15075,17404,20032,861,3416,57440,41872,32548,29776,32548,41872,57440,3416,13140,155904,
+A229755 ,0,0,0,1,3,1,3,60,60,3,12,422,598,422,12,50,1840,2347,2347,1840,50,210,6456,6809,6561,6809,6456,210,861,20032,17404,15075,15075,17404,20032,861,3416,57440,41872,32548,29776,32548,41872,57440,3416,13140,155904,97565,69198,57677,57677,69198,97565,155904,13140,
A229756 ,2,2,4,2,12,6,2,32,28,8,2,82,110,48,10,2,206,408,224,72,12,2,516,1454,968,378,100,14,2,1294,5048,4016,1784,578,132,16,2,3252,17244,16202,7980,2924,830,168,18,2,8194,58290,64058,34570,13810,4464,1140,208,20,
A229757 ,1,2,3,4,5,7,8,9,11,12,15,17,20,21,23,29,36,39,41,44,84,
A229758 ,0,0,0,0,0,0,0,0,9,9,0,9,9,18,18,0,9,9,18,27,27,0,9,18,18,27,36,36,0,9,18,27,36,36,45,54,0,9,18,27,36,45,54,63,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,9,9,9,18,0,0,0,
@@ -232849,7 +232849,7 @@ A232844 ,988,500516,179149960,32297599754,10331973738132,3491302545600310,101368
A232845 ,0,1,0,-2,-4,-4,4,44,236,1300,8276,61484,523804,5024036,53478980,624890236,7946278604,109195935284,1612048228276,25439293045580,427278358483196,7609502950269124,143217213477235364,2840152418116021916,59189357288576068780,
A232846 ,1,2,4,6,3,8,9,10,12,5,14,15,16,18,20,21,7,22,24,25,11,26,27,28,30,32,33,34,35,13,36,38,39,40,42,44,45,46,48,17,49,50,51,52,54,55,56,57,58,60,19,62,63,64,65,66,68,69,70,72,74,75,76,23,77,78,80,81,
A232847 ,1,22,17310,20802,23110,24262,25995,26542,29427,31735,33835,38137,39287,39859,40967,13595040,14285160,15129504,15378336,15834528,15912936,16327008,16555752,16897896,16908264,17054388,17145432,17749044,18013428,20239146,20713482,21265578,
-A232848 ,2,59,97,127,12517,54581,83921,89273,1396411,2562719,4952183,29201281,35562101,47567557,111213143,184201627,1172476337,7309217299,287609314877,
+A232848 ,2,59,97,127,12517,54581,83921,89273,1396411,2562719,4952183,29201281,35562101,47567557,111213143,184201627,1172476337,7309217299,287609314877,5173838081669,
A232849 ,29,112,356,960,2307,5078,10265,19632,35694,62317,104971,170736,269815,415810,625923,923930,1339958,1911980,2686162,3721083,5089904,6879942,9196707,12169730,15946892,20711810,26672412,34082534,43239305,54478386,
A232850 ,112,1169,10020,69096,395858,1977447,8733769,35038672,129401308,446455554,1449355171,4445398493,12943299647,35960503702,95710408560,245007196179,
A232851 ,356,10020,250841,5049004,81165760,1100553095,12972830143,135388726387,1274013956481,10999910504747,
@@ -233556,7 +233556,7 @@ A233551 ,419,659,1769,2609,2651,2981,
A233552 ,361,919,1681,1849,2419,2629,3301,5209,5539,5581,6421,7771,8551,9109,9871,10039,10609,10819,11491,13399,13729,13771,14611,15661,15961,16741,17299,18061,18229,18799,19009,19681,21589,21919,21961,22801,24151,24931,25489,
A233553 ,53542288800,59509850400,59999219280,60074174160,61695597600,67154527440,68763895200,69626138400,71957405520,72598125600,67509842400,72747675000,73605331800,75710489400,78953074200,87113426400,88410722400,89398663200,96058282320,96369633360,
A233554 ,1,3,6,15,19,22,207,542,2374,10579,17726,43182,
-A233555 ,2,5724469,10534369,16784723,33330911,189781037,8418091991,58605633953,109388266843,448366797199,1056238372873,24603683667221,86982253895059,100316149840769,
+A233555 ,2,5724469,10534369,16784723,33330911,189781037,8418091991,58605633953,109388266843,448366797199,1056238372873,24603683667221,86982253895059,100316149840769,164029709175817,
A233556 ,1,2,4,6,10,12,116,147,324,2070,2902,3663,4994,11531,13554,22421,558905,1242890,1655487,2021278,2878297,4790338,7061177,16875261,21813642,24563860,58919808,69676102,85356321,92610708,205600836,338430087,343675600,1176903461,1698127637,4657254361,17421656611,
A233557 ,2,3,7,13,29,37,641,853,2143,18059,26417,34283,48539,122597,146539,254831,8304757,19534651,26528699,32820527,47825363,82199141,124088207,312168289,409464961,464174839,1167927947,1393486043,1725361103,1879982849,4346448019,7331901341,7451088943,27036461983,39662532977,113692593373,449281234057,
A233558 ,1,2,2,3,0,3,4,1,1,4,5,2,3,2,5,6,3,5,5,3,6,7,4,7,0,7,4,7,8,5,0,2,2,0,5,8,9,6,1,4,5,4,1,6,9,10,7,2,6,8,8,6,2,7,10,11,8,3,8,11,0,11,8,3,8,11,12,9,4,10,1,3,3,1,10,4,9,12,13,10,5,12,3,6,7,6,3,12,5,10,13,14,11,6,14,5,9,11,11,9,5,14,
@@ -233768,7 +233768,7 @@ A233763 ,0,1,2,2,4,2,4,5,8,2,4,6,12,6,8,11,16,2,4,6,12,14,16,18,24,10,8,14,28,14
A233764 ,0,1,3,5,9,11,15,21,29,31,35,41,51,61,69,83,99,101,105,111,121,131,141,159,183,201,209,223,245,271,287,317,349,351,355,361,371,381,391,409,433,451,461,479,507,545,575,625,679,713,721,735,757,783,
A233765 ,0,1,2,2,4,2,4,6,8,2,4,6,10,10,8,14,16,2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32,2,4,6,10,10,10,18,24,18,10,18,28,38,30,50,54,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64,2,4,6,10,10,
A233766 ,2,9,6,7,2,9,3,7,9,7,2,9,3,2,7,6,7,9,3,2,7,9,6,2,2,9,6,7,2,9,3,7,9,7,2,9,3,2,7,6,7,9,3,2,7,9,6,2,2,9,6,7,2,9,3,7,9,7,2,9,3,2,7,6,7,9,3,2,7,9,6,2,2,9,6,7,2,9,3,7,9,7,2,9,3,2,7,6,7,9,3,2,7,9,6,2,
-A233767 ,2,97,3203,5059,6469,8081,35051,39719,42209,109049,154591,523297,6621827,20059771,258196441,731584957,1427109029,1899496631,8428550519,50790885203,7475902096387,22626378502139,38855796912367,
+A233767 ,2,97,3203,5059,6469,8081,35051,39719,42209,109049,154591,523297,6621827,20059771,258196441,731584957,1427109029,1899496631,8428550519,50790885203,7475902096387,22626378502139,38855796912367,162082298018497,
A233768 ,1,2,4,5,6,10,12,53,226,361,400,620,935,1037,3832,3960,4956,7222,12183,13615,24437,80849,450827,680044,7388490,23503578,27723887,52048944,85860268,126177976,606788411,613917734,2693408896,3856356590,5167833600,5810025660,9197308014,10805855623,19751202045,19781610414,27240188169,30742119459,
A233769 ,2,3,7,11,13,29,37,241,1429,2437,2741,4583,7333,8269,36073,37397,48121,73037,130261,147289,280037,1032259,6594787,10249573,130193849,443038781,527454197,1024907927,1736090963,2602512709,13517865841,13684220029,64209198247,93380481511,126718347859,143176188581,231059158871,273286859737,511940464493,512760363097,715173864563,810985955573,
A233770 ,2,7,6,8,5,7,6,2,4,8,6,2,5,7,6,5,3,8,9,3,6,4,3,7,2,5,0,8,2,3,5,7,3,3,9,6,3,1,7,9,7,9,7,3,7,5,2,7,5,1,3,7,3,9,1,5,9,7,7,3,1,6,4,3,5,4,8,5,0,1,4,1,8,0,8,2,9,7,1,2,4,3,1,1,8,9,8,
@@ -245301,7 +245301,7 @@ A245296 ,1,0,4,4,2,5,7,9,0,9,3,0,9,7,9,5,1,4,3,4,4,5,3,6,9,6,1,7,1,5,5,7,0,2,5,8
A245297 ,1,1,1,6,6,4,5,9,7,1,1,0,3,8,0,9,8,8,2,6,4,5,7,1,5,4,5,1,0,7,3,1,5,3,1,7,8,9,6,6,5,1,2,0,0,6,6,9,7,4,0,4,0,1,6,4,5,6,3,4,2,1,6,0,6,0,8,1,7,9,5,2,8,6,4,8,5,2,2,2,9,6,8,4,6,4,6,0,0,2,6,2,2,4,5,4,9,9,1,2,3,
A245298 ,1,1,1,9,4,2,3,7,3,1,7,3,5,1,0,7,6,1,1,6,2,9,7,1,1,0,8,2,0,8,1,2,6,1,0,4,1,2,4,9,9,8,5,5,6,7,0,5,8,6,0,7,0,8,6,5,2,0,9,8,2,7,9,9,1,3,1,5,4,2,2,9,2,2,9,6,9,0,4,5,1,5,2,5,2,6,2,8,6,5,9,6,1,3,0,8,5,2,2,9,2,9,5,2,
A245299 ,1,4,9,6,2,7,7,8,6,9,7,3,8,8,4,4,7,3,8,5,0,8,1,0,2,1,3,9,3,2,9,7,8,2,5,5,3,3,1,7,0,0,6,2,4,7,0,9,3,2,5,4,1,0,3,0,8,7,5,6,8,6,3,9,5,0,3,6,8,0,0,9,7,2,0,4,5,0,0,4,3,3,7,4,5,7,0,3,5,8,1,0,9,0,8,3,9,6,3,9,6,9,2,0,9,
-A245300 ,0,1,4,3,7,12,6,11,17,24,10,16,23,31,40,15,22,30,39,49,60,21,29,38,48,59,71,84,28,37,47,58,70,83,97,112,36,46,57,69,82,96,111,127,144,45,56,68,81,95,110,126,143,161,180,55,67,80,94,109,125,142,160,
+A245300 ,0,1,4,3,7,12,6,11,17,24,10,16,23,31,40,15,22,30,39,49,60,21,29,38,48,59,71,84,28,37,47,58,70,83,97,112,36,46,57,69,82,96,111,127,144,45,56,68,81,95,110,126,143,161,180,55,67,80,94,109,125,142,160,179,199,220,
A245301 ,0,5,22,58,120,215,350,532,768,1065,1430,1870,2392,3003,3710,4520,5440,6477,7638,8930,10360,11935,13662,15548,17600,19825,22230,24822,27608,30595,33790,37200,40832,44693,48790,53130,57720,62567,67678,73060,78720,84665,
A245302 ,3,5,9,13,19,25,32,39,48,57,67,78,90,103,116,130,145,161,178,195,213,232,252,273,294,317,340,364,388,414,440,467,495,524,554,584,615,647,680,714,748,783,820,856,894,933,972,1012,1053,1095,1137,1181,1225,1270,1316,1362,
A245303 ,2,3,5,7,8,11,12,13,16,17,18,19,20,23,24,27,28,29,31,32,37,40,41,43,44,45,47,48,50,52,53,54,56,59,61,63,64,67,68,71,72,73,75,76,79,80,81,83,88,89,92,96,97,98,99,101,103,104,107,108,109,112,113,116,117,124,125,127,128,131,
@@ -261175,7 +261175,7 @@ A261170 ,2,3,4,5,7,9,17,20,31,38,43,64,64,70,91,93,102,117,120,123,127,127,127,1
A261171 ,2,3,4,4,5,6,9,10,13,16,16,21,23,23,29,28,38,39,33,34,41,40,37,37,41,42,44,64,77,82,75,83,83,87,104,104,86,94,
A261172 ,2,3,2,4,3,6,9,10,11,16,12,14,22,18,25,20,2,6,18,14,7,40,31,25,23,20,22,62,65,68,29,23,38,26,104,6,34,52,
A261173 ,11,0,101,0,0,0,0,10111,0,0,0,101111,0,0,0,0,0,0,1011001,0,0,0,11110111,0,10011101,10010101,0,0,0,101111111,101101111,0,100100111,101001001,0,0,0,0,1010111111,1001110111,0,1000011011,1000001011,0,0,
-A261174 ,1,2,9,30,90,248,650,1560,3560,7680,15786,31076,58905,107768,191180,329664,554038,909558,1461655,2302950,3563482,5422392,8124040,11997648,17482295,25156872,
+A261174 ,1,2,9,30,90,248,650,1560,3560,7680,15786,31076,58905,107768,191180,329664,554038,909558,1461655,2302950,3563482,5422392,8124040,11997648,17482295,25156872,35779092,50330364,70072640,96615760,131999058,178786960,240186182,320179470,
A261175 ,1,1,1,3,5,8,13,19,26,35,45,56,69,84,100,117,137,158,180,204,231,258,288,319,352,387,424,463,503,546,590,636,684,734,786,840,897,955,1015,1077,1141,1207,1275,1345,1418,1492,1568,1647,1728,1811,1896,1983,2072,2163,2257,2352,
A261176 ,0,9,126,802,3158,10040,25464,58837,123422,238203,429467,733923,1200319,1912928,2945116,4369570,6338678,9053512,12622814,17359779,23503546,31347788,41161317,
A261177 ,0,10,180,1392,6149,21350,57192,137617,298864,593378,1101739,1936342,3216080,
@@ -262699,7 +262699,7 @@ A262694 ,0,0,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7
A262695 ,0,4,0,3,2,2,0,1,1,24,3,23,0,1,2,22,2,21,0,1,1,20,0,19,1,1,3,18,1,17,0,16,2,1,0,15,1,1,10,14,1,2,0,1,2,13,0,12,9,1,1,11,1,10,0,1,1,9,0,8,8,7,0,1,1,6,1,1,1,5,0,4,7,3,1,1,13,2,0,1,2,12,4,11,6,1,3,10,1,5,0,9,2,4,0,8,5,7,1,3,1,2,0,1,4,6,0,5,1,1,2,4,1,1,0,3,1,1,0,2,3,
A262696 ,0,2,0,1,1,1,0,1,1,13,1,13,0,1,1,11,1,11,0,1,1,10,0,10,1,1,1,10,1,9,0,8,1,1,0,8,1,1,6,7,1,1,0,1,1,6,0,6,5,1,1,6,1,5,0,1,1,5,0,3,4,3,0,1,1,3,1,1,1,2,0,1,4,1,1,1,7,1,0,1,1,7,1,6,4,1,1,6,1,1,0,5,1,1,0,4,4,4,1,1,1,1,0,1,3,4,0,4,1,1,1,3,1,1,0,1,1,1,0,1,3,0,4,1,
A262697 ,0,6,0,3,2,2,0,1,1,38,3,37,0,1,2,33,2,32,0,1,1,30,0,29,1,1,3,28,1,26,0,24,2,1,0,23,1,1,16,21,1,2,0,1,2,18,0,17,13,1,1,16,1,14,0,1,1,13,0,10,11,9,0,1,1,8,1,1,1,6,0,4,10,3,1,1,23,2,0,1,2,22,4,20,9,1,3,19,1,5,0,13,2,4,0,11,8,10,1,3,1,2,0,1,6,9,0,8,1,1,2,6,1,1,0,3,1,1,0,2,5,0,12,1,
-A262698 ,1,2,4,24,41,51,88,95,99,179,183,663,782,829,1339,2054,2816,7918,8474,13264,16664,27415,39514,48606,51145,
+A262698 ,1,2,4,24,41,51,88,95,99,179,183,663,782,829,1339,2054,2816,7918,8474,13264,16664,27415,39514,48606,51145,187222,200906,261980,353209,375162,396967,400469,
A262699 ,1,2,2,3,4,4,8,8,16,16,32,32,64,64,128,128,256,256,512,512,1024,1024,2048,2048,4096,4096,8192,8192,16384,16384,32768,32768,65536,65536,131072,131072,262144,262144,524288,524288,1048576,1048576,
A262700 ,5,19,31,151,691,1181,1489,1511,1601,2579,3037,7297,9661,10993,11699,20407,25657,33937,65099,96419,102911,133157,251789,411841,417271,670729,808211,1179907,1671277,
A262701 ,9,6,3,9,7,2,3,8,4,4,0,2,1,9,4,1,0,5,2,7,1,1,4,5,9,2,6,2,3,6,4,8,2,3,1,5,6,2,6,7,2,8,9,5,2,5,8,2,1,9,0,6,4,5,6,1,0,9,5,7,9,7,0,0,5,6,4,0,3,5,6,4,7,8,6,3,3,7,0,3,9,0,7,2,2,8,7,3,1,6,5,0,0,8,7,9,6,7,8,8,8,3,1,1,5,
@@ -262723,7 +262723,7 @@ A262718 ,0,0,2,18,194,2550,39962,730002,15257090,359376750,9424209002,2723850294
A262719 ,1,6,21,55,110,203,357,544,808,1177,1670,2215,2865,3599,4558,5621,6637,8041,9769,11413,13394,15593,17683,20317,23249,26063,29506,33287,37461,41692,46306,50707,55667,61723,67547,73939,80767,87941,94913,101613,111422,
A262720 ,1,2,8,22,68,198,586,1718,5047,14808,43470,127636,374957,1102078,3241082,9537070,28079357,82718212,243809138,718994032,2121378272,6262089436,18493519148,54639914652,161503493023,477558890378,1412658185320,
A262721 ,1,11,14,1215,1811,111211,1419,2215,1120,1116,1811,111211,1419,2215,1120,1116,1811,111211,1419,2215,1120,1116,1811,111211,1419,2215,1120,1116,1811,111211,1419,2215,1120,1116,1811,111211,1419,2215,1120,1116,
-A262722 ,1,41,56,74,103,157,384,491,537,868,1490,1710,
+A262722 ,1,41,56,74,103,157,384,491,537,868,1490,1710,4322,4523,4877,4942,5147,5407,7564,17576,67722,
A262723 ,105,231,627,897,935,1581,1729,2465,2967,4123,4301,4715,5487,7685,7881,9717,10707,11339,14993,16377,17353,20213,20915,23779,25327,26331,26765,29341,29607,32021,33335,40587,40807,42911,48635,49321,54739,55581,55637,59563,60297,63017,
A262724 ,1,3,10,28,36,91,1081,2278,2926,8001,46665,5639761,10911456,166066200,341532180,3137785371,1647882316985625,875366737297292691171,465198187808352499674075441,
A262725 ,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,
@@ -263617,7 +263617,7 @@ A263612 ,0,1,4,121,10201,12321,114411,1002001,1234321,100020001,102030201,121242
A263613 ,0,1,1331,1030301,1003003001,1000300030001,1000030000300001,1000003000003000001,1000000300000030000001,1000000030000000300000001,1000000003000000003000000001,1000000000300000000030000000001,1000000000030000000000300000000001,1000000000003000000000003000000000001,
A263614 ,0,0,1,2,2,4,4,8,8,16,15,30,26,52,42,84,64,128,93,186,130,260,176,352,232,464,299,598,378,756,470,940,576,1152,697,1394,834,1668,988,1976,1160,2320,1351,2702,1562,3124,1794,3588,2048,4096,2325,4650,2626,5252,2952,5904,3304,6608,3683,7366,
A263615 ,2,4,8,12,20,28,44,59,89,115,167,209,293,357,485,578,764,894,1154,1330,1682,1914,2378,2677,3275,3653,4409,4879,5819,6395,7547,8244,9638,10472,12140,13128,15104,16264,18584,19935,22637,24199,27323,29117,32705,34753,38849,41174,45824,48450,
-A263616 ,4,3,8,5,11,6,19,14,25,18,
+A263616 ,4,3,8,5,11,6,19,14,25,18,49,
A263617 ,4,7,15,20,31,37,56,70,95,113,
A263618 ,4,0,3,0,7,1,5,0,11,0,5,1,19,0,13,1,25,0,18,0,
A263619 ,4,4,7,7,14,15,20,20,31,31,36,37,56,56,69,70,95,95,113,113,
@@ -276134,7 +276134,7 @@ A276129 ,1,3,6,13,27,54,106,204,387,725,1344,2469,4500,8145,14652,26213,46665,82
A276130 ,1,1,1,1,1,4,10,55,649,38881,6414706,24978826228,2913605221297249,112139525368095766797655,8403341152380185679389503620974065,146904111947501701959735285821948223340424963459227,
A276131 ,1,1,1,1,1,10,136,20251,413100001,170660037240000001,2912484838126132813026335712191641,62371823031725048177115183368983888882661870237372850050710016335,
A276132 ,2,3,5,7,11,13,17,31,37,71,73,79,97,131,151,157,179,337,353,359,373,727,733,739,751,757,929,937,953,971,1733,1979,3319,3371,3373,3719,3733,7177,7717,9133,9173,9791,
-A276133 ,0,2,1,4,2,5,1,3,6,1,8,4,1,3,7,6,2,8,3,3,4,5,6,9,3,1,4,2,5,11,8,6,1,10,1,6,7,3,6,6,2,8,6,3,1,12,10,6,2,4,4,4,8,11,4,6,1,7,4,1,11,13,3,3,3,15,7,8,2,6,4,7,7,5,3,10,7,5,7,
+A276133 ,0,2,1,4,2,5,1,3,6,1,8,4,1,3,7,5,2,8,3,3,4,5,6,9,3,1,4,2,5,11,8,6,1,10,1,6,7,3,6,6,2,8,6,3,1,12,10,6,2,4,4,4,8,11,4,6,1,7,4,1,11,13,3,3,3,15,7,8,2,6,4,7,7,5,3,10,7,5,7,
A276134 ,0,1,1,1,1,1,2,2,2,2,1,2,2,2,2,1,2,2,2,2,1,2,2,2,2,1,2,2,2,2,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,1,2,2,2,2,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,1,2,2,2,2,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,1,2,2,2,2,2,3,3,3,3,2,3,3,3,3,2,3,3,3,3,2,
A276135 ,0,0,1,20,51,2604,6665,720600,1864135,348678440,909090909,261535698060,685853880635,281241170407092,740800455037201,410525522232055664,1085102592571150095,781282469559318055056,2070863582910344082917,1879498672877297909667780,4993219047619047619047619,5577014881186619679500164220,
A276136 ,6,11,12,13,23,47,192,193,383,786432,
@@ -277588,7 +277588,7 @@ A277583 ,1,4,1,10,5,2,4,14,1,12,3,2,3,9,1,31,2,1,15,7,5,6,2,3,12,20,1,19,11,2,2,
A277584 ,0,1,25,784,27225,1002001,38291344,1502337600,60101954649,2440703175625,100300325150025,4161829109817600,174077451630810000,7330421677037621904,310467090932230849600,13214837914326197526784,564927069263895118093401,
A277585 ,1,3,15,21,315,3465,45045,15015,765765,14549535,14549535,25741485,1673196525,1003917915,145568097675,265447707525,1504203675975,4512611027925,166966608033225,33393321606645,1369126185872445,58872425992515135,294362129962575675,
A277586 ,1,4,22,32,488,5408,70544,23552,1202048,22846976,22850816,40431616,2628156416,1576923136,228655904768,416962576384,2362792902656,7088385949696,262270410489856,52454094798848,2150618140770304,92476585387491328,462382939977023488,
-A277587 ,0,0,1,1,9,9,7,3,2,2,2,3,
+A277587 ,0,0,1,1,9,9,7,3,2,2,2,3,1,0,3,2,8,8,
A277588 ,1,10,11,21,31,41,51,61,71,81,91,100,101,110,111,121,131,141,151,161,171,181,191,201,210,211,221,231,241,251,261,271,281,291,301,310,311,321,331,341,351,361,371,381,391,401,410,411,421,431,441,451,461,471,
A277589 ,2,12,20,22,32,42,52,62,72,82,92,102,112,120,122,132,142,152,162,172,182,192,200,202,212,220,222,232,242,252,262,272,282,292,302,312,320,322,332,342,352,362,372,382,392,402,412,420,422,432,442,452,462,472,
A277590 ,3,13,23,30,33,43,53,63,73,83,93,103,113,123,130,133,143,153,163,173,183,193,203,213,223,230,233,243,253,263,273,283,293,300,303,313,323,330,333,343,353,363,373,383,393,403,413,423,430,433,443,453,463,473,
@@ -280927,7 +280927,7 @@ A280922 ,2,24,1744,769408,2063048448,33639061257216,3336558889746769920,20135476
A280923 ,4,16,80,768,9536,223232,6867200,393936896,29989282816,4225123221504,795427838939136,275571189819113472,128240735455510216704,109332361699222156738560,125729867860804073988096000,263919716304200619134696816640,749827702212803707621023160729600,3876699219598969046471294814225694720,
A280924 ,0,2,3,4,14,27,35,42,53,60,89,117,126,137,162,207,281,472,2752,3381,6462,12183,14910,29205,40883,50675,78717,83880,99113,
A280925 ,1,13,25,29,33,46,57,61,129,187,676,779,828,1621,1666,1693,2237,2249,2872,3993,5148,6190,6457,25019,30358,60100,
-A280926 ,5,7,29,47,119,699,1407,4911,18971,46803,119951,363209,
+A280926 ,5,7,29,47,119,699,1407,4911,18971,46803,119951,363209,1276197,3722389,19973297,73605289,183273481,390720475,1671075265,4541314567,22107473795,44810965685,172567099183,617945607281,1835952288687,3938674815741,19847928172101,
A280927 ,1735,2469,4341,4569,4989,5469,5637,5961,6879,7149,7407,8675,9969,11569,12949,13057,13089,13707,15829,15969,16407,18597,18969,19959,20109,20487,20721,21081,21309,21729,22107,22221,22513,23469,24355,25269,25617,26305,27021,
A280928 ,1255,12955,17482,25105,100255,101299,105295,107329,117067,124483,127417,129595,132565,145273,146137,149782,163797,174082,174298,174793,174982,250105,256315,263155,295105,297463,307183,325615,371893,536539,687919,1002955,1004251,1012099,1025095,1029955,
A280929 ,2,3,2,1,4,1,2,2,3,1,2,2,4,1,2,3,3,4,2,2,6,1,4,3,3,1,2,2,4,3,2,4,6,1,2,5,5,1,2,3,3,7,8,2,4,1,2,4,3,4,6,2,6,3,2,3,3,1,2,4,4,1,17,5,6,7,2,2,5,3,2,6,5,1,2,2,4,3,4,4,3,1,6,8,8,1,2,3,3,5,2,2,6,1,15,5,3,4,2,5,
@@ -282445,7 +282445,7 @@ A282440 ,0,1276,378380,28043694,2301995900,177265299282,12904221882656,926611650
A282441 ,0,0,0,0,0,0,0,2,2,0,0,12,96,12,0,0,58,784,784,58,0,0,280,6498,10232,6498,280,0,0,1276,50962,152726,152726,50962,1276,0,0,5592,378380,2129756,3997136,2129756,378380,5592,0,0,24004,2744000,28043694,98841792,
A282442 ,2,3,3,4,6,5,5,9,9,8,10,11,11,15,15,11,12,18,19,16,20,17,15,24,25,18,20,28,19,24,26,21,21,31,31,20,28,25,21,32,40,33,31,39,39,25,25,35,35,51,47,32,40,54,55,48,50,41,39,60,59,58,63,59,49,50,58,
A282443 ,1,2,2,3,5,4,4,8,8,7,9,10,10,14,14,10,11,17,18,15,19,16,14,23,24,17,19,27,18,23,25,20,20,30,30,19,27,24,20,31,39,32,30,38,38,24,24,34,34,50,46,31,39,53,54,47,49,40,38,59,58,57,62,58,48,49,57,39,
-A282444 ,1,2,5,8,14,50,119,200,269,299,1154,5369,
+A282444 ,1,2,5,8,14,50,119,200,269,299,1154,5369,47249,48299,58643,130325,148979,282074,887480,
A282445 ,4,3,3,3,4,3,4,3,4,3,7,3,12,6,8,4,13,7,8,4,11,3,20,5,6,22,11,23,13,16,14,9,10,10,24,29,6,40,31,0,3,4,40,11,32,45,13,7,30,3,53,20,6,30,35,27,54,26,0,63,46,57,16,67,67,38,0,39,52,5,61,75,3,
A282446 ,1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,4,2,6,2,6,4,4,2,6,3,4,3,6,2,8,2,3,4,4,4,9,2,4,4,6,2,8,2,6,6,4,2,8,3,6,4,6,2,6,4,6,4,4,2,12,2,4,6,5,4,8,2,6,4,8,2,9,2,4,6,6,4,8,2,8,4,4,2,12,4,4,
A282447 ,1,0,111,100,11011,11110,1110011,1010110,110101111,111011000,11101111111,10111000000,1101101111111,1111111000000,111000001111111,101011101000000,11010110110011111,11101111111010000,1110111000001100111,1011101011111100100,
@@ -282975,7 +282975,7 @@ A282970 ,1,0,1,0,1,1,1,1,1,1,2,1,2,2,2,3,2,4,3,4,4,4,5,5,5,6,6,7,7,8,9,9,10,10,1
A282971 ,1,0,1,0,1,1,1,2,1,3,2,4,4,6,7,9,11,15,18,24,29,37,48,58,78,92,124,149,195,243,308,393,490,629,786,1004,1263,1603,2024,2564,3239,4106,5184,6571,8301,10508,13298,16807,21296,26895,34082,43060,54528,68952,87245,110392,139622,176696,223484,282798,357731,
A282972 ,1,1,1,1,2,2,3,1,2,4,3,2,2,4,4,2,2,3,5,2,2,4,4,2,3,3,3,2,2,3,2,2,1,4,2,1,4,2,3,1,4,3,2,1,3,5,2,1,3,6,3,2,2,5,5,2,4,3,4,2,3,5,2,2,2,6,5,2,4,5,6,1,5,6,5,4,5,5,6,2,4,
A282973 ,3,31,314159,314159265359,
-A282974 ,1,2,6,12,1902,3971,5827,
+A282974 ,1,2,6,12,1902,3971,5827,16208,47577,
A282975 ,0,1,6,13,18,28,40,45,50,70,101,210,248,298,1246,1340,1586,2466,6548,6713,7394,23904,32450,38171,39120,67816,108610,112400,129038,
A282976 ,1,0,1,0,111,101,10000,111,1110011,1010001,100100100,1,11111111100,10000100101,1000110000000,10110111111,111000101000011,101010000011011,10010000111010101,110110000010,1111110110110111000,1000010011100001011,100011000011101100111,
A282977 ,1,0,100,0,11100,101000,100,11100000,110011100,1000101000,100100100,100000000000,11111111100,10100100001000,11000100,1111110110100000,11000010100011100,110110000010101000,1010101110000100100,1000001101100000000,111011011011111100,
@@ -286397,7 +286397,7 @@ A286392 ,1,6,231,1284066,352654485156,3553786240466361696,1289303099816839265917
A286393 ,1,7,406,5105212,4154189102413,167633579843887699759,331466355732596931093508048522,32115447190132359991237336502881651018804,152470060954479462517322396167243320349298407119379801,
A286394 ,1,8,666,16912512,35184646816768,4722366500530551259136,40564819207305653446303190876160,22300745198530623151211847196048401987796992,784637716923335095479473759060307277562325323313332617216,
A286395 ,1,3,7,8,9,11,15,19,29,55,76,159,266,311,394,908,1732,1875,4335,6334,7641,16421,33721,139239,157705,160143,
-A286396 ,1,9,1035,48700845,231628411446741,89737248564744874067889,2816049943117424212512789695666175,7158021121277935153545945911617993395398302485,
+A286396 ,1,9,1035,48700845,231628411446741,89737248564744874067889,2816049943117424212512789695666175,7158021121277935153545945911617993395398302485,1473773072217322896440109113309952350877179744639518847951721,
A286397 ,1,10,1540,125512750,1250002537502500,1250000000501250002500000,125000000000000250375000000250000000,1250000000000000000005001250000000002500000000000,
A286398 ,1,7,143,7429,94395,70514711,68421139647,3628781953225,180465781280744001,1051696554978819009,2043771643161196817,455757414124192757820663,145129235359794615466069,1358004768744860147421669766123,9043798410819212324167588503127725,
A286399 ,0,0,1,8,32,96,244,528,1024,1856,3126,5016,7808,11616,16808,23856,32768,44352,59293,77352,100032,128128,161052,201264,249856,305280,371294,450128,537856,640992,762744,894528,1048576,1228224,1419858,1642080,1897376,2167008,
@@ -290789,7 +290789,7 @@ A290784 ,1,3,6,19,213,379687,80990506,22635546606,
A290785 ,2,16,150,5771,270411,51462132,
A290786 ,1,1,-1,-23,3429,8425506,-412878084725,-497641562809372379,17436260499054618815283977,20503694883570579788445502041773422,-917439693541287252616828116888122637934368489,-1746281566732870051764961051797990328294109372786185933382,
A290787 ,1,8,44,83,265,378,58,267,783,2890,289,5802,6781,9866,12390,15274,4288,9223,22764,30890,6595,42130,49725,58010,1575,76770,87305,7670,110835,123890,53786,127309,168575,11048,10389,1884,164216,116326,86857,188924,73351,15241,30690,81318,45139,157378,511828,41849,594784,638890,
-A290788 ,6,56,656,8656,38656,238656,7238656,47238656,447238656,
+A290788 ,6,56,656,8656,38656,238656,7238656,47238656,447238656,7447238656,27447238656,227447238656,3227447238656,
A290789 ,1,1,1,1,1,1,1,1,0,1,1,1,-1,-1,1,1,1,-2,-7,0,1,1,1,-3,-23,47,2,1,1,1,-4,-55,586,873,0,1,1,1,-5,-109,3429,48778,-26433,-5,1,1,1,-6,-191,13436,885137,-11759396,-1749159,0,1,1,1,-7,-307,40915,8425506,-904638963,-8596478231,220526159,14,1,
A290790 ,1,6,32,590,21555,1598353,
A290791 ,6,9,16,27,28,95,96,121,122,123,124,125,126,537,538,539,540,905,906,1149,1150,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,9585,9586,15719,15720,15721,15722,15723,15724,15725,15726,19653,19654,19655,
@@ -301739,7 +301739,7 @@ A301734 ,1,-1,1,0,-1,1,1,0,-1,0,2,1,1,-1,-2,0,2,3,2,1,0,-3,-2,1,4,4,5,
A301735 ,1,-1,-2,-2,-2,0,1,5,7,11,13,16,14,14,8,0,-12,-26,-46,-66,-90,-114,-135,-155,-169,
A301736 ,1,0,1,1,7,11,281,449,20719,75403,3066769,1234967,821856311,2223747371,273942958057,1238828105761,12489209350781,511763293389419,13479473195610647,356089289643109313,78908612931754624999,373825489242185563339,83933730864756536571961,
A301737 ,1,1,2,3,8,30,144,840,5760,45360,403200,570240,43545600,518918400,6706022400,93405312000,126804787200,22230464256000,53801459712000,6758061133824000,128047474114560000,2554547108585472000,53523844179886080000,
-A301738 ,3,3,3,5,3,3,3,49,7,35,67,75,157,107,71,137,275,
+A301738 ,3,3,3,5,3,3,3,49,7,35,67,75,157,107,71,137,275,531,
A301739 ,2,4,10,17,30,44,67,91,126,163,213,265,333,403,491,582,693,807,944,1084,1249,1418,1614,1814,2044,2278,2544,2815,3120,3430,3777,4129,4520,4917,5355,5799,6287,6781,7321,7868,8463,9065,9718,10378,11091,11812,12588,13372,14214,15064,
A301740 ,3,9,24,50,96,164,267,408,603,856,1186,1598,2115,2742,3505,4411,5489,6746,8215,9904,11849,14059,16573,19401,22586,26138,30103,34493,39357,44707,50596,57037,64086,71757,80109,89157,98964,109545,120966,133244,146448,160595,175758,191955,
A301741 ,1,2,10,76,778,10026,155884,2839880,59339004,1399069450,36746349496,1064024248068,33676500286840,1156685567791586,42850609041047760,1703182952266379536,72299420602524921616,3264579136056004359570,156238968782480840396704,7900247992586138688381500,
@@ -304658,7 +304658,7 @@ A304653 ,1,-1,-1,0,-1,1,-1,0,0,1,-1,-1,-1,1,1,0,-1,-1,-1,-1,1,1,-1,1,0,1,0,-1,-1
A304654 ,0,0,4,27,328,6500,192216,7952112,438941952,31185057024,2772643115520,301622403456000,39413353102848000,6091955683706880000,1099401414283210752000,229088914497045356544000,54589580461769879715840000,14750581694440372638842880000,
A304655 ,0,0,8,81,2480,175000,23825904,5563712448,2051674085376,1124193889529856,873600549068759040,927968580453961728000,1307864687259363065856000,2386263863328126193631232000,5521179117888960788194394112000,15917227342113559040727019683840000,
A304656 ,5,4,4,1,3,9,8,0,9,2,7,0,2,6,5,3,5,5,1,7,8,2,2,3,4,7,7,2,9,2,6,4,6,7,1,9,6,8,5,2,1,9,8,7,4,4,2,7,8,2,2,1,7,2,6,7,0,9,6,5,4,8,0,6,1,6,4,3,6,9,5,4,3,3,7,9,0,6,1,6,5,1,0,5,2,3,7,4,9,6,4,6,3,6,1,8,
-A304657 ,604,176,11008,1460,176,35392,176,604,51648,1888,176,107552,176,176,51648,1460,15488,179712,604,1460,132928,176,1460,312896,7880,3172,211728,604,176,179712,176,1460,132928,1460,604,586688,1888,1460,132928,1460,176,468352,176,
+A304657 ,26,76,76,208,176,252,176,544,208,604,176,768,176,604,604,1376,176,768,176,1888,604,604,176,2208,818,604,544,1888,176,2316,176,3392,604,604,1460,2568,176,604,604,5536,176,2316,176,1888,1888,604,176,6080,818,3172,
A304658 ,7,3,5,4,6,7,0,6,2,6,0,1,2,2,4,1,4,5,9,3,3,0,7,2,6,3,3,0,9,6,4,8,4,7,7,3,7,7,4,3,7,6,9,7,0,6,8,6,3,8,8,0,4,5,5,3,7,3,9,3,9,3,0,8,9,2,3,2,2,2,0,6,8,9,3,0,0,3,2,0,3,9,3,1,7,1,2,
A304659 ,0,5,31,94,210,395,665,1036,1524,2145,2915,3850,4966,6279,7805,9560,11560,13821,16359,19190,22330,25795,29601,33764,38300,43225,48555,54306,60494,67135,74245,81840,89936,98549,107695,117390,127650,138491,149929,161980,174660,187985,
A304660 ,1,2,4,6,8,18,16,30,36,54,32,150,64,162,108,210,128,450,256,750,324,486,512,1470,216,1458,900,3750,1024,2250,2048,2310,972,4374,648,7350,4096,13122,2916,10290,8192,11250,16384,18750,4500,39366,32768,25410,1296,
@@ -306886,7 +306886,7 @@ A306881 ,0,1,4,18,120,1100,13092,192360,3362128,68063760,1565077220,40275499264,
A306882 ,22,34,38,46,58,62,76,78,82,86,92,98,102,106,118,122,138,142,152,154,158,164,166,172,178,182,190,194,202,212,214,218,226,238,244,254,258,262,266,274,278,282,298,302,304,310,316,318,322,328,332,334,338,344,346,356,358,362,
A306883 ,5,9,3,2,3,7,2,9,7,7,6,9,7,2,8,4,6,4,5,5,2,0,6,0,1,9,7,9,4,7,0,8,1,7,0,0,4,2,3,8,8,3,8,8,2,3,6,2,1,6,5,7,7,7,4,5,7,6,7,1,2,8,6,0,9,6,9,9,5,0,5,8,7,1,0,6,7,8,5,7,9,1,5,2,9,0,7,1,4,3,3,3,5,3,7,9,4,8,8,4,9,4,3,2,5,4,3,5,1,
A306884 ,1,1,3,6,14,28,93,270,86170,7625640881546,
-A306885 ,1,1,1,0,2,5,6,17,1,6,1,19,6,2,2,
+A306885 ,1,1,1,0,2,5,6,17,1,6,1,19,6,2,2,10,26,
A306886 ,0,5,14,26,41,60,82,105,134,164,197,234,272,314,359,407,456,507,566,623,686,748,812,883,956,1030,1107,1181,1267,1354,1445,1529,1620,1721,1814,1920,2022,2121,2232,2344,2460,2573,2691,2815,2936,3061,3189,3321,3462,
A306887 ,0,6,396,200100,1368937020,
A306888 ,0,1,1,2,1,4,3,8,11,20,31,64,105,202,367,696,1285,2452,4599,8776,16651,31838,60787,116640,223697,430396,828525,1598228,3085465,5966000,11545611,22371000,43383571,84217616,163617805,318150720,619094385,1205614054,2349384031,4581315968,
@@ -309109,7 +309109,7 @@ A309104 ,0,1,3,9,25,72,199,545,1487,4048,11007,29930,81371,221199,601295,1634499
A309105 ,1,1,3,9,25,71,198,543,1486,4045,11007,29931,81371,221197,601294,1634497,4443046,12077467,32829975,89241140,242582583,659407855,1792456409,4872401706,13244561047,36002449653,97864804698,266024120284,723128532126,1965667148553,
A309106 ,1,1,1,1,2,2,1,3,6,2,1,4,12,16,4,1,5,20,44,10,2,1,6,30,96,90,36,6,1,7,42,174,240,84,28,4,1,8,56,288,690,336,168,48,6,1,9,72,440,1344,984,336,144,36,4,1,10,90,640,2590,3060,2100,1200,450,100,10,
A309107 ,0,0,0,2,0,2,2,3,0,4,0,2,5,0,3,7,0,3,3,4,10,0,5,10,3,6,0,5,5,6,4,11,0,6,4,4,5,8,0,6,6,7,26,0,5,8,8,9,0,5,5,6,11,21,0,6,4,21,4,2,48,0,7,21,6,9,18,0,6,4,11,18,5,22,0,7,13,0,3,54,0,3,3,4,14,0,5,14,3,6,21,27,0,7,18,23,0,4,14,11,
-A309108 ,1,1,1,2,3,2,5,6,7,4,10,9,7,11,12,8,13,11,17,19,15,23,7,14,16,12,10,13,29,16,31,25,27,37,26,19,33,31,41,34,35,18,43,41,47,40,32,43,25,23,53,29,9,59,21,43,51,52,53,38,28,49,54,58,
+A309108 ,1,1,1,2,3,2,5,6,7,4,10,9,7,11,12,8,13,11,17,19,15,23,7,14,16,12,27,13,25,29,31,37,33,30,26,16,20,27,34,29,35,18,41,43,47,53,39,37,49,51,59,38,40,41,46,47,42,19,31,44,55,56,61,57,67,64,45,71,62,
A309109 ,1,1,933120,2681795837952000,237391215092234044047360000000,647223519675870437718855767650467840000000000000,254101032901646255941392101056649724780871931658240000000000000000000,
A309110 ,1,81,75582720,217225462874112000,19228688422470957567836160000000,52425105093745505455227317179687895040000000000000,20582183665033346731252760185588627707250626464317440000000000000000000,
A309111 ,1,1,2009078326886400,25130033447370922318407480728239472640000000,5759627596191312699511553760965199283079808523515804251057792885981184000000000000000,
@@ -327361,8 +327361,8 @@ A327356 ,0,0,1,3,40,1365,
A327357 ,1,0,1,1,1,4,1,3,1,30,13,33,32,6,546,421,1302,1915,1510,693,316,135,45,10,1,
A327358 ,1,1,0,2,1,0,5,3,2,0,20,14,10,6,0,180,157,128,91,54,0,
A327359 ,1,1,0,1,1,0,2,1,2,0,6,4,4,6,0,23,29,37,37,54,0,
-A327360 ,3,44,355,3195,99733,833719,5419351,80143857,657408909,
-A327361 ,1,14,113,1017,31746,265381,1725033,25510582,209259755,
+A327360 ,3,44,355,3195,99733,833719,5419351,80143857,657408909,6167950454,42106686282,983339177173,8958937768937,94960529682104,428224593349304,6134899525417045,66627445592888887,430010946591069243,5293386250278608690,31760317501671652140,
+A327361 ,1,14,113,1017,31746,265381,1725033,25510582,209259755,1963319607,13402974518,313006581566,2851718461558,30226875395063,136308121570117,1952799169684491,21208174623389167,136876735467187340,1684937174853026414,10109623049118158484,
A327362 ,0,0,1,3,28,475,14646,813813,82060392,15251272983,5312295240010,3519126783483377,4487168285715524124,11116496280631563128723,53887232400918561791887118,513757147287101157620965656285,9668878162669182924093580075565776,
A327363 ,1,1,0,2,1,0,8,4,1,0,64,38,10,1,0,1024,728,238,26,1,0,
A327364 ,0,0,1,6,46,655,17991,927416,89009740,16020407709,5468601546685,3578414666656214,4529751815161579194,11175105490563109463875,54043272967471942825421219,514566625051705610110588073460,9677104749727084630538798805505880,
@@ -331388,7 +331388,7 @@ A331383 ,0,0,0,0,0,0,1,0,2,2,1,1,1,2,2,2,1,4,2,2,2,4,2,3,4,1,3,4,5,0,3,3,1,6,2,1
A331384 ,35,65,95,98,154,324,364,476,623,763,791,812,826,938,994,1036,1064,1106,1144,1148,1162,1288,1484,1708,1736,2044,2408,2632,4320,5408,6688,6974,8000,10208,12623,12701,12779,14144,19624,23144,25784,26048,44176,47696,
A331385 ,1,0,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,0,2,3,1,1,0,0,0,1,4,3,1,2,0,0,0,0,2,5,3,2,2,0,1,0,0,0,0,1,4,6,3,4,2,0,2,0,0,0,0,0,2,6,6,4,6,2,1,2,0,1,0,0,0,0,0,1,4,8,6,6,7,2,4,2,0,1,0,0,0,1,
A331386 ,3,5,6,9,10,11,12,15,17,18,20,21,22,24,25,27,30,31,33,34,35,36,39,40,41,42,44,45,48,50,51,54,55,57,59,60,62,63,65,66,67,68,69,70,72,75,77,78,80,81,82,83,84,85,87,88,90,93,95,96,99,100,102,105,108,
-A331387 ,1,2,4,7,11,16,24,34,47,64,86,113,148,191,245,310,390,486,602,740,907,1104,1338,1613,1937,2315,2758,3272,3871,4562,5362,
+A331387 ,1,2,4,7,11,16,24,34,47,64,86,113,148,191,245,310,390,486,602,740,907,1104,1338,1613,1937,2315,2758,3272,3871,4562,5362,6283,7344,8558,9952,11542,13356,15419,17766,20425,23440,26846,30696,35032,39917,45406,
A331388 ,1,0,2,3,9,3,20,12,24,10,54,15,77,21,48,48,135,24,170,57,103,55,252,60,240,78,216,123,405,47,464,192,273,136,390,144,665,171,388,228,819,102,902,327,456,253,1080,240,1008,240,678,465,1377,216,1036,492,853,406,1710,
A331389 ,1,1,3,29,666,28344,1935054,193926796,26892165502,4946464286746,1168900475263013,346080409272270888,125798338606148948325,55204084562033205121607,28834556615453989801860765,17710828268156331289770544579,12658784968736373972502731143309,
A331390 ,1,9,29,68,134,237,388,600,887,1265,1751,2364,3124,4053,5174,6512,8093,9945,12097,14580,17426,20669,24344,28488,33139,38337,44123,50540,57632,65445,74026,83424,93689,104873,117029,130212,144478,159885,176492,194360,213551,
@@ -332725,7 +332725,7 @@ A332720 ,1,1,5,19,59,150,349,745,1515,2936,5514,10036,17851,31039,53006,88943,14
A332721 ,1,1,3,72,5752,1501620,1171326960,2571831080160,15245263511750160,236246829658682027760,9325247205993698149853760,917699267902161951609308035200,221117091698491444413008381486903040,128433050637127079872089064922773889126400,
A332722 ,1,1,2,9,74,711,7312,77793,848557,9426039,106218592,1210785512,13933358426,161624712815,1887635428421,22176331059637,261881397819259,3106736469937751,37006306302036790,442425926101676831,5306994321265281854,63851605555921588684,770371217568310624912,
A332723 ,1,2,1,3,4,4,10,0,1,5,19,3,3,6,31,13,6,7,46,35,10,8,65,74,14,9,92,131,18,10,140,192,27,1,11,202,274,46,3,12,275,396,62,3,13,363,563,79,9,14,467,784,100,14,15,598,1054,126,12,2,
-A332724 ,0,0,1,6,14,32,65,128,243,
+A332724 ,0,0,1,6,14,32,65,128,243,452,826,1490,2659,4704,8261,14418,25030,43252,74437,127648,218199,371920,632306,1072486,1815239,3066432,5170825,8705118,14632958,24562952,41177801,68947520,115313979,192656924,321554986,536191418,
A332725 ,90,126,180,198,234,252,270,306,342,350,360,378,396,414,450,468,504,522,525,540,550,558,594,612,630,650,666,684,700,702,720,738,756,774,792,810,825,828,846,850,882,900,910,918,936,950,954,975,990,1008,1026,1044,
A332726 ,1,1,2,4,8,16,31,61,120,228,438,836,1580,2976,5596,10440,19444,36099,66784,123215,226846,416502,763255,1395952,2548444,4644578,8452200,15358445,27871024,50514295,91446810,165365589,298730375,539127705,972099072,1751284617,3152475368,
A332727 ,0,0,0,0,0,0,1,3,8,28,74,188,468,1120,2596,5944,13324,29437,64288,138929,297442,632074,1333897,2798352,5840164,12132638,25102232,51750419,106346704,217921161,445424102,908376235,1848753273,3755839591,7617835520,15428584567,31207263000,
@@ -332875,7 +332875,7 @@ A332870 ,0,0,0,0,0,0,2,9,32,92,243,587,1361,3027,6564,13928,29127,60180,123300,2
A332871 ,0,0,0,0,1,4,8,24,55,128,282,625,1336,2855,6000,12551,26022,53744,110361,225914,460756,937413,1902370,3853445,7791647,15732468,31725191,63907437,128613224,258626480,519700800,1043690354,2094882574,4202903667,8428794336,16897836060,
A332872 ,1,1,3,10,34,116,396,1352,4616,15760,
A332873 ,0,0,0,0,22,340,3954,44716,536858,
-A332874 ,0,0,0,0,0,0,0,0,0,0,10,10,20,30,50,150,180,290,420,630,860,
+A332874 ,0,0,0,0,0,0,0,0,0,0,10,10,20,30,50,150,180,290,420,630,860,1828,2168,3326,4514,6530,8576,12188,20096,25314,35576,48062,65592,86752,117222,152060,237590,292346,402798,524596,711270,910606,1221204,1554382,2044460,2927124,
A332875 ,3,3,3,3,3,4,2,3,3,3,3,3,3,3,3,2,3,4,3,3,3,3,3,3,3,2,4,3,3,3,3,3,4,2,3,3,3,3,3,3,3,4,2,3,4,3,3,3,2,4,3,2,3,4,3,3,3,2,3,4,2,3,3,3,3,3,2,4,3,3,3,3,3,3,3,3,4,3,2,3,3,3,4,2,3,4,3,
A332876 ,12,14,36,28,105,102,147,136,108,120,242,204,286,238,330,352,374,306,2109,140,462,484,2047,408,150,572,594,756,3219,360,682,864,2937,1326,770,792,4107,2128,4329,280,3649,1638,3827,1232,990,2530,5217,1344,5439,1050,
A332877 ,6,15,21,35,55,77,91,143,187,221,253,323,391,493,551,667,713,899,1073,1189,1271,1517,1591,1763,1961,2183,2419,2537,2773,3127,3233,3599,3953,4189,4331,4757,4897,5293,5723,5963,6499,6887,7171,7663,8051,8633,8989,9797,9991,10403,10807,11303,
@@ -333194,7 +333194,7 @@ A333189 ,0,0,0,0,1,0,1,0,0,1,1,0,1,1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0
A333190 ,1,1,2,2,4,5,7,10,13,15,21,26,29,39,49,50,68,80,92,109,129,142,181,201,227,262,317,343,404,456,516,589,677,742,870,949,1077,1207,1385,1510,1704,1895,2123,2352,2649,2877,3261,3571,3966,4363,4873,5300,5914,6466,
A333191 ,1,1,2,2,5,8,10,18,24,29,44,60,68,100,130,148,201,256,310,396,478,582,736,898,1068,1301,1594,1902,2288,2750,3262,3910,4638,5510,6538,7686,9069,10670,12560,14728,17170,20090,23462,27292,31710,36878,42704,49430,
A333192 ,1,1,2,2,4,5,7,10,14,16,24,31,37,51,67,76,103,129,158,199,242,293,370,450,538,652,799,953,1147,1376,1635,1956,2322,2757,3271,3845,4539,5336,6282,7366,8589,10046,11735,13647,15858,18442,21354,24716,28630,32985,
-A333193 ,1,1,2,3,5,7,11,15,21,29,40,53,71,93,122,158,204,260,332,419,528,
+A333193 ,1,1,2,3,5,7,11,15,21,29,40,53,71,93,122,158,204,260,332,419,528,661,825,1023,1267,1560,1916,2344,2860,3476,4217,5097,6147,7393,8872,10618,12685,15115,17977,21336,25276,29882,35271,41551,48872,57385,67277,78745,92040,
A333194 ,1,2,4,4,8,8,11,11,19,16,21,21,30,30,37,29,45,45,51,51,66,56,67,67,88,83,96,84,105,105,112,112,144,130,147,135,159,159,178,162,197,197,208,208,241,209,232,232,277,270,290,270,309,309,324,308,357,335,364,364,
A333195 ,8,16,24,27,30,32,40,48,54,56,60,64,72,80,81,88,96,104,105,108,110,112,120,125,128,135,136,144,150,152,160,162,168,176,184,189,192,200,208,210,216,220,224,232,238,240,243,248,250,256,264,270,272,273,280,288,
A333196 ,1,2,6,6,30,10,70,70,210,210,2310,2310,30030,30030,30030,30030,510510,510510,9699690,1939938,646646,646646,14872858,44618574,223092870,223092870,223092870,223092870,6469693230,6469693230,200560490130,200560490130,18232771830,
@@ -335460,7 +335460,7 @@ A335455 ,0,0,0,1,1,5,11,30,69,142,334,740,1526,3273,6840,14251,29029,59729,12200
A335456 ,1,2,5,12,32,84,211,556,1446,3750,9824,25837,67681,178160,468941,1233837,3248788,
A335457 ,1,2,5,12,31,80,196,486,1171,2787,6564,15323,35403,81251,185087,418918,942525,2109143,4695648,10405694,22959156,
A335458 ,1,2,2,3,2,3,3,4,2,3,3,5,3,5,5,5,2,3,3,5,3,5,5,7,3,5,5,8,5,8,7,6,2,3,3,5,3,4,5,7,3,5,4,7,5,7,8,9,3,5,5,8,4,8,7,11,5,8,7,11,7,11,9,7,2,3,3,5,3,4,5,7,3,5,5,7,5,7,8,9,3,5,5,8,5,7,
-A335459 ,0,0,0,0,4,18,102,786,3960,51450,675570,10804710,139674024,2793377664,
+A335459 ,0,0,0,0,4,18,102,786,3960,51450,675570,10804710,139674024,2793377664,58662908640,1798893694080,26985313555200,782574083010720,25992638958686400,857757034323189000,30021498596590300800,1563341714743040232000,64179292280096037844800,2631350957341279888915200,
A335460 ,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,2,0,0,0,1,1,0,0,3,0,1,0,1,0,2,0,2,0,0,0,6,0,0,1,0,0,0,0,1,0,0,0,8,0,0,1,1,0,0,0,3,0,0,0,6,0,0,0,
A335461 ,1,0,1,0,1,2,0,1,4,8,0,1,6,24,44,0,1,8,48,176,308,0,1,10,80,440,1540,2612,0,1,12,120,880,4620,15672,25988,0,1,14,168,1540,10780,54852,181916,296564,0,1,16,224,2464,21560,146272,727664,2372512,3816548,
A335462 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
@@ -337514,7 +337514,7 @@ A337513 ,1,-1,0,1,0,-1,-5,13,5,-43,4,98,122,-638,-246,2912,-537,-9419,-1648,4700
A337514 ,1,-1,0,1,0,-2,1,-1,13,-16,-39,76,122,-365,-64,537,1103,-1565,-6850,6630,38704,-58273,-108054,204722,366920,-598506,-1526994,1111475,9656314,-7254090,-43224847,39704799,171028427,-177129071,-604754108,
A337515 ,1,1,1,4,1,2,1,8,4,21,29,
A337516 ,1,2,4,9,23,57,151,421,1202,3397,9498,25970,70005,187259,500061,
-A337517 ,1,1,2,4,9,23,57,151,427,1263,3823,11724,36048,110953,342079,1064468,3341067,10583564,
+A337517 ,1,1,2,4,9,23,57,151,427,1263,3823,11724,36048,110953,342079,1064468,3341067,10583564,33727683,
A337518 ,1,1,2,1,2,1,0,0,1,0,2,2,0,0,1,2,0,1,0,2,1,2,0,0,2,0,0,1,1,1,0,0,2,0,1,0,2,1,0,0,2,2,0,2,0,0,2,1,2,1,0,1,0,1,1,1,0,1,2,2,1,0,1,2,0,1,2,1,1,1,1,0,0,2,1,1,2,0,2,0,0,0,2,1,2,2,1,
A337519 ,4,15,28,47,68,95,124,159,196,239,284,335,388,447,508,575,644,719,796,879,964,1055,1148,1247,1348,1455,1564,1679,1796,1919,2044,2175,2308,2447,2588,2735,2884,3039,3196,3359,3524,3695,3868,4047,4228,4415,4604,4799,4996,
A337520 ,1,1,2,4,10,22,64,147,409,1092,3253,8661,28585,83190,274001,912373,3366384,13253582,61533277,290493694,
@@ -337999,8 +337999,8 @@ A337998 ,2,3,9,9,6,3,5,2,4,4,9,5,6,3,0,9,5,5,3,3,7,5,7,4,3,1,0,1,6,0,5,7,7,2,2,5
A337999 ,2,2,0,0,11,33,29,60,3905,19524,62879,275436,10187165,71191608,481719419,3211782240,101121160145,904977244224,10099756468559,89733565369536,2746252055597525,29900664884062848,479479605967022099,5296351543857279360,166991194742961246905,
A338000 ,0,0,2,4,1,1,151,604,1135,3652,163921,983020,4781635,26455096,880441381,7019296864,62338135855,485246558272,14909515819441,147911335595200,2005509679122475,19997668777814656,618177354753297901,7327199316870984064,135962126415847073095,
A338001 ,1,0,1,0,2,2,0,6,2,3,0,24,8,4,3,4,0,120,8,12,6,6,4,5,0,720,48,16,48,18,6,18,8,8,5,6,0,5040,48,48,240,18,24,12,72,12,8,24,10,10,6,7,0,40320,384,96,192,1440,36,36,24,36,360,32,12,32,16,96,15,10,30,12,12,7,8,
-A338002 ,56,816,6064,18152,52088,100608,208168,
-A338003 ,37,653,5517,17153,50349,97037,204329,
+A338002 ,56,816,6064,18152,52088,100608,208168,336840,579136,846560,1310960,1784888,
+A338003 ,37,653,5517,17153,50349,97037,204329,330613,571021,835713,1298533,1764125,
A338004 ,6,6,3,4,8,2,9,7,0,5,1,1,4,3,4,8,0,8,0,5,7,5,6,8,8,4,7,4,3,7,2,3,9,9,5,0,0,0,5,0,4,2,8,9,8,5,1,5,6,9,6,2,5,5,4,5,7,1,8,2,4,4,9,9,5,0,5,9,3,3,1,5,0,9,3,7,7,6,8,3,8,5,0,6,8,1,0,9,7,9,1,5,6,8,7,8,5,8,9,8,7,3,3,3,0,1,0,9,0,8,3,3,8,9,1,3,9,4,5,4,
A338005 ,1,2,16,2592,466308864,
A338006 ,1,2,3,3,4,5,6,6,7,7,8,8,9,9,10,10,11,11,12,13,13,13,14,14,15,15,15,16,17,17,18,18,19,19,19,19,20,20,20,21,22,23,24,24,24,24,25,25,25,25,26,26,27,27,28,28,29,29,30,30,31,31,31,31,31,31,32,32,33,33,34,34,35,35,36,36,36,36,37,37,37,37,38,
@@ -338193,7 +338193,7 @@ A338193 ,1,1,2,10,100,1556,33016,888952,29035280,1115554960,49300214176,24638594
A338194 ,1,1,4,42,828,24840,1009440,51906960,3232993680,236644571520,19911894206400,1893868822137600,200939416407576000,23530201619699174400,3014512836056949427200,419416309548107359488000,62979130153042151656608000,10151678353264190993682432000,
A338195 ,18,20,35,36,40,45,54,56,60,70,72,77,80,84,90,100,104,105,108,110,112,120,126,135,140,143,144,154,160,162,168,170,175,176,180,182,189,198,200,208,209,210,216,220,221,224,225,231,234,240,245,252,260,264,266,270,
A338196 ,1,2,4,8,3,5,6,10,12,16,20,24,32,13,26,40,48,7,9,11,14,17,18,22,28,34,36,44,52,56,21,42,64,68,72,80,84,19,25,29,38,50,58,76,88,96,33,66,100,104,37,74,112,116,128,132,45,90,136,144,49,98,148,152,15,
-A338197 ,1,2,4,8,20,44,114,312,894,2639,8005,24362,74918,231123,722388,2276599,7242497,
+A338197 ,1,2,4,8,20,44,114,312,894,2639,8005,24362,74918,231123,722388,2276599,7242497,23144119,
A338198 ,1,0,1,2,1,1,2,3,2,1,6,5,4,3,1,10,11,8,5,4,1,22,21,16,11,6,5,1,42,43,32,21,14,7,6,1,86,85,64,43,26,17,8,7,1,170,171,128,85,54,31,20,9,8,1,342,341,256,171,106,65,36,23,10,9,1,682,683,512,341,214,127,76,41,26,11,10,1,
A338199 ,1,1,3,1,1,1,2,2,1,3,5,1,1,1,2,3,1,1,3,1,1,1,2,2,1,2,4,1,1,2,2,4,1,1,3,1,1,3,2,2,1,5,7,1,1,1,2,3,1,1,3,1,1,1,2,2,1,2,4,1,1,3,2,5,1,1,3,1,1,1,2,2,1,
A338200 ,0,0,1,2,4,6,9,12,17,21,27,33,41,48,58,67,79,90,104,117,134,149,168,186,208,228,253,276,304,330,361,390,425,457,495,531,573,612,658,701,751,798,852,903,962,1017,1080,1140,1208,1272,1345,1414,1492,1566,1649,
@@ -338466,6 +338466,7 @@ A338466 ,0,1,2,3,4,5,6,7,8,9,10,12,11,13,14,15,16,19,18,17,20,21,22,23,26,24,27,
A338467 ,1,3,4,7,8,13,12,19,16,25,24,29,32,35,36,41,44,49,48,57,54,61,62,67,70,77,76,81,82,85,88,101,94,109,98,121,102,129,110,135,118,143,122,155,126,161,130,175,144,181,148,187,156,191,168,199,176,207,180,215,
A338468 ,15,33,35,51,55,69,77,85,93,95,105,119,123,141,143,145,155,161,165,177,187,195,201,205,209,215,217,219,221,231,249,253,255,265,285,287,291,295,309,323,327,329,335,341,345,355,357,381,385,391,395,403,407,411,
A338469 ,125,275,425,575,605,775,935,1025,1175,1265,1331,1445,1475,1675,1705,1825,1955,2057,2075,2255,2425,2575,2585,2635,2645,2725,2783,3175,3179,3245,3425,3485,3565,3685,3725,3751,3925,3995,4015,4175,4301,4475,4565,4715,
+A338470 ,1,0,0,0,0,1,0,3,2,5,5,13,7,23,21,33,35,65,55,104,97,151,166,252,235,377,399,549,591,846,858,1237,1311,1749,1934,2556,2705,3659,3991,5090,5608,7244,7841,10086,11075,13794,15420,19195,21003,26240,29089,35483,
A338471 ,8,20,44,50,68,92,110,124,125,164,170,188,230,236,242,268,275,292,310,332,374,388,410,412,425,436,470,506,508,548,575,578,590,596,605,628,668,670,682,716,730,764,775,782,788,830,844,902,908,932,935,964,970,
A338472 ,3,109,14519,2024592291,1536463613637,2449395996564189425,4686662617019462175259,33724155827962966577589860263,2606282943971359343146382147809434583605,15159042500551578738018590862773479717960671,6576976543997974825092367662248938303820921894460988333,
A338473 ,8,10,12,13,14,15,16,18,19,20,21,23,24,25,27,28,29,30,31,34,35,36,39,40,41,42,44,45,46,49,50,53,55,56,58,59,60,63,64,70,74,84,98,125,127,130,131,135,136,142,146,147,149,152,153,156,157,158,164,168,170,
@@ -338794,12 +338795,12 @@ A338857 ,1,8,15,23,30,38,45,52,60,67,74,82,89,97,104,111,119,126,134,141,148,156
A338858 ,2,1,0,9,3,2,9,9,2,7,6,2,0,0,4,9,1,8,9,3,9,1,9,5,2,8,6,4,0,2,1,5,6,5,7,6,7,5,9,2,1,1,1,5,3,8,5,1,7,3,2,6,1,1,0,1,9,3,7,8,4,7,9,5,0,1,8,8,6,4,2,0,7,6,8,4,7,2,6,6,2,1,6,0,2,0,8,8,8,6,3,9,3,6,0,0,2,1,0,6,6,4,1,9,8,
A338859 ,1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,4,3,1,1,0,1,9,10,4,1,1,0,1,20,45,20,6,1,1,0,1,48,210,165,55,8,1,1,0,1,115,1176,1540,1035,136,13,1,1,0,1,286,6670,19600,22155,6273,430,18,1,1,0,1,719,41041,260130,692076,324008,46185,1300,30,1,1,0,
A338860 ,0,1,0,2,1,3,4,6,8,11,17,21,30,38,53,68,90,115,150,192,243,312,390,496,613,775,951,1193,1456,1810,2200,2715,3285,4026,4856,5909,7106,8595,10301,12394,14809,17728,21118,25171,29891,35489,42018,49702,58678,69180,
-A338861 ,1,2,6,15,42,143,399,1190,4209,10920,37245,109886,339745,1037186,3205734,9784263,29837784,93313919,
+A338861 ,1,2,6,15,42,143,399,1190,4209,10920,37245,109886,339745,1037186,3205734,9784263,29837784,93313919,289627536,
A338862 ,1,1,4,13,49,175,655,2437,9208,34867,132952,508621,1953580,7524625,29061835,112493680,436330753,1695388480,6598016866,25714222228,100343852938,392023844362,1533182752336,6001993189687,23517048084424,92220047277892,361906295452669,1421252193947311,
A338863 ,1,1,2,3,4,5,1,1,2,3,4,5,2,2,4,6,8,10,3,3,6,9,12,15,4,4,8,12,16,20,5,5,10,15,20,25,1,1,2,3,4,5,1,1,2,3,4,5,2,2,4,6,8,10,3,3,6,9,12,15,4,4,8,12,16,20,5,5,10,15,20,25,2,2,4,6,8,10,2,2,4,
A338864 ,1,4,1,12,12,1,72,96,24,1,240,840,360,40,1,2880,7200,4920,960,60,1,10080,70560,65520,19320,2100,84,1,161280,745920,887040,362880,58800,4032,112,1,1088640,7983360,12640320,6652800,1481760,150192,7056,144,1,
A338865 ,1,6,1,24,18,1,168,204,36,1,720,2280,780,60,1,8640,25200,14400,2100,90,1,40320,292320,252000,58800,4620,126,1,604800,3729600,4334400,1486800,183120,8904,168,1,4717440,46811520,76265280,35743680,6335280,474768,15624,216,1,
-A338866 ,0,0,0,0,0,0,4,5,18,65,267,1238,6196,33480,187932,1095882,6629220,
+A338866 ,0,0,0,0,0,0,4,5,18,65,267,1238,6196,33480,187932,1095882,6629232,
A338867 ,1,1,5,38,424,6284,
A338868 ,0,0,0,0,0,0,0,0,0,0,0,2,3,9,28,138,613,2798,
A338869 ,1,1,2,2,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,30,30,30,30,6,30,6,6,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,30,
@@ -339468,7 +339469,7 @@ A339544 ,3,17,19,29,31,71,79,83,103,113,151,211,229,293,331,337,347,349,421,439,
A339545 ,3,19,29,691,
A339546 ,1,1,1,1,1,1,1,1,1,1,1,1,2,3,6,6,15,17,40,45,89,116,199,271,
A339547 ,15,172,1114,5378,22321,83995,293744,968965,
-A339548 ,2,3,4,7,11,19,35,56,105,177,321,610,1001,1893,3186,5714,
+A339548 ,2,3,4,7,11,19,35,56,105,177,321,610,1001,1893,3186,5714,10073,
A339549 ,1,1,2,1,2,4,3,1,4,4,3,8,3,9,16,1,2,16,3,8,18,9,4,16,6,9,16,27,4,256,5,1,12,4,18,64,3,9,24,16,3,324,4,27,128,16,5,32,9,36,16,27,4,256,30,81,24,16,5,4096,5,25,216,1,12,144,3,8,24,324,4,256,3,
A339550 ,1,9,85,697,1285,2605,4573,5845,6001,6241,6613,7141,7453,8005,10897,12453,13141,15445,19789,20345,21445,21913,22873,25957,36565,36601,39597,44761,46405,53677,56137,56593,61013,63445,70094,72913,76977,80913,82405,87085,87601,
A339551 ,513059433,3007912105,4791685641,11555664153,44615854297,111890605585,121111724905,163901238153,
@@ -339482,6 +339483,8 @@ A339558 ,0,1,1,1,0,3,0,1,2,1,0,3,0,1,2,1,0,4,0,1,2,1,0,3,0,1,2,1,0,5,0,1,1,1,0,5
A339559 ,0,0,1,0,2,1,4,3,7,6,14,14,23,27,41,47,70,84,114,141,190,225,303,370,475,578,738,890,1131,1368,1698,2058,2549,3048,3759,4505,5495,6574,7966,9483,11450,
A339560 ,1,0,0,1,1,2,2,4,5,8,8,13,17,22,28,39,48,62,81,101,127,167,202,253,318,395,486,608,736,906,1113,1353,1637,2011,2409,2922,3510,4227,5060,6089,7242,
A339561 ,1,6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,60,62,65,69,74,77,82,84,85,86,87,90,91,93,94,95,106,111,115,118,119,122,123,126,129,132,133,134,140,141,142,143,145,146,150,155,156,158,159,161,166,
+A339562 ,1,15,33,35,51,55,69,77,85,91,93,95,105,119,123,141,143,145,155,161,165,177,187,195,201,203,205,209,215,217,219,221,231,247,249,253,255,265,285,287,291,295,299,301,309,323,327,329,335,341,345,355,357,377,381,
+A339563 ,2,3,5,6,7,10,11,13,14,17,19,21,22,23,26,29,30,31,34,37,38,39,41,42,43,46,47,53,57,58,59,61,62,65,66,67,70,71,73,74,78,79,82,83,86,87,89,94,97,101,102,103,106,107,109,110,111,113,114,115,118,122,127,
A339565 ,1,3,17,101,627,3999,25955,170571,1131433,7559301,50795985,342935689,2324278669,15804931797,107775401349,736723618773,5046774983235,34636814325087,238114193665451,1639378334244867,11301978856210543,78010917772099207,539055832175992119,
A339566 ,5,137,3967,25087,242899421,
A339567 ,1,5,15,25,55,91,137,525,625,925,3967,5995,7625,10767,25087,57225,68817,565027,591415,2515825,2757625,4162019,5276309,96689255,115686005,133890625,242899421,492029715,588620625,1839399055,7786281065,11231388063,17251448809,71050380625,
@@ -339657,6 +339660,7 @@ A339742 ,1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,4,1,0,2,2,2,1
A339743 ,1,1,2,4,6,6,6,30,30,60,60,210,210,210,210,210,210,210,2310,2310,2310,2310,18480,120120,120120,150150,150150,150150,150150,660660,1531530,2492490,3063060,3063060,4594590,38798760,38798760,38798760,38798760,38798760,48498450,193993800,
A339744 ,4,8,9,16,18,24,25,27,32,36,48,49,54,64,72,80,81,96,100,108,112,121,125,128,135,144,160,162,169,192,196,200,216,224,225,243,250,256,288,289,320,324,343,352,360,361,375,384,392,400,405,416,432,441,448,450,480,484,486,500,
A339745 ,9,9,9,0,0,5,4,4,2,4,8,0,9,8,9,4,7,5,2,7,3,7,8,4,5,3,5,8,5,4,2,2,7,2,4,5,8,6,0,5,9,0,9,7,3,8,5,3,6,4,7,3,6,9,0,8,2,2,8,9,6,2,3,9,9,2,8,9,5,9,9,4,1,9,5,9,8,9,8,1,0,0,7,4,1,1,8,6,0,3,5,0,2,7,7,3,1,7,1,3,0,5,0,9,0,6,
+A339746 ,1,5,6,7,8,11,13,17,19,23,25,27,29,30,31,35,36,37,40,41,42,43,47,48,49,53,55,56,59,61,64,65,66,67,71,73,77,78,79,83,85,88,89,91,95,97,101,102,103,104,107,109,113,114,115,119,121,125,127,131,133,135,
A339747 ,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,31,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,31,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,31,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,6,1,1,1,1,31,
A339748 ,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,43,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,43,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,1,7,1,1,1,1,
A339749 ,2,3,2,4,2,7,2,3,2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2,4,2,9,2,3,2,4,2,8,2,3,2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2,4,2,11,2,3,2,4,2,8,2,3,2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2,
@@ -339667,6 +339671,7 @@ A339753 ,1,2,3,11,23,24,29,31,108,109,198,199,240,241,243,244,245,246,247,248,24
A339754 ,1,0,2,0,2,3,0,2,6,6,0,4,12,16,10,0,8,24,40,40,20,0,20,60,104,120,90,35,0,50,150,270,350,330,210,70,0,140,420,768,1040,1080,840,448,126,0,392,1176,2184,3080,3468,3108,2128,1008,252,
A339755 ,1,2,5,11,27,55,131,263,571,1168,2445,4891,10113,20227,40979,82229,165632,331265,665365,1330731,2666729,5334769,10679319,21358639,42740683,85482096,171004645,342015001,684113793,1368227587,2736633741,5473267483,10946869669,21893763789,43788190107,
A339756 ,1,4,4,8,4,17,12,15,14,33,12,58,28,43,52,113,39,140,57,124,129,240,66,241,173,270,217,362,58,388,292,454,351,539,166,783,471,723,463,880,229,1134,642,843,763,1441,311,1415,740,1295,987,1888,357,1629,1063,1750,1231,2381,289,2652,
+A339757 ,9,1,2,1,1,1,1,2,3,0,1,2,3,1,0,0,1,3,2,0,1,2,3,0,0,0,1,2,2,0,
A339758 ,3,3,53,503,4297,947,10589,17903,624401,7151083,45543077,30611047,612126937,2280521251,649288301,26566080479,28921314337,303937208923,1086758949557,12299159511127,39118361784041,18314722943123,64249761922429,2484777068103119,1148475719438129,14810825716436683,
A339759 ,1,2,4,6,30,60,210,2310,18480,120120,150150,660660,1531530,2492490,3063060,4594590,38798760,48498450,193993800,446185740,6915878970,13831757940,80313433200,129393864600,1061029689720,5014012253250,9225782545980,12033629407800,40312658516130,135378330837750,
A339760 ,1,12,48,208,768,2752,9472,32000,106496,351232,1150976,3756032,12222464,39698432,128778240,417398784,1352138752,4378591232,14175698944,45886734336,148520304640,480679821312,1555633799168,5034389536768,16292153131008,52723609239552,170619454881792,552140862914560,
@@ -339676,6 +339681,7 @@ A339763 ,1,768,43676,4743130,364618672,28808442502,2125185542510,153198148096800
A339764 ,6,9,7,2,6,5,6,5,1,0,7,7,0,3,2,0,8,9,2,3,3,3,9,8,4,3,7,5,0,0,0,0,0,0,0,0,2,5,8,6,0,8,7,5,7,1,8,0,9,0,3,2,5,1,7,5,3,1,2,2,0,3,8,1,8,8,8,9,4,0,4,9,1,2,0,1,0,6,4,2,2,4,8,9,8,5,9,2,5,4,7,3,1,9,2,5,3,7,5,3,8,1,2,5,1,7,9,7,0,8,0,0,3,9,9,7,8,0,2,7,3,4,3,7,5,0,0,0,0,0,0,
A339765 ,-1,0,-1,0,1,0,1,0,1,2,1,2,3,2,3,2,3,4,3,4,3,4,5,4,5,6,5,6,5,6,7,6,7,8,7,8,7,8,9,8,9,8,9,10,9,10,11,10,11,10,11,12,11,12,11,12,13,12,13,14,13,14,13,14,15,14,15,16,15,16,15,16,17,16,17,16,
A339766 ,2,6,1,2,0,0,0,7,4,0,4,3,4,5,2,6,0,6,4,4,3,7,3,7,1,1,3,0,9,5,4,4,5,6,7,2,4,3,3,4,0,4,5,8,7,3,7,0,9,3,8,2,6,6,0,9,3,5,1,0,8,0,6,0,5,1,5,6,0,4,1,0,8,8,7,4,9,3,0,1,3,6,2,5,1,3,6,
+A339767 ,2,3,0,5,1,7,-2,0,3,11,-1,13,5,2,-4,17,-2,19,1,4,9,23,-3,0,11,-3,3,29,0,31,-6,8,15,2,-4,37,17,10,-1,41,2,43,7,-1,21,47,-5,0,-2,14,9,53,-5,6,1,16,27,59,-2,61,29,1,-8,8,6,67,13,20,0,71,-6,73,
A339768 ,1,1,1,1,1,1,1,1,3,1,1,1,5,25,1,1,1,7,109,543,1,1,1,9,289,9449,29281,1,1,1,11,601,63487,3068281,3781503,1,1,1,13,1081,267249,69711361,3586048685,1138779265,1,
A339769 ,1,2,4,4,5,6,10,12,9,7,4,9,13,11,7,6,8,10,13,14,10,15,14,21,8,7,13,21,
A339770 ,1,15,170,2766,46127,811265,14605298,268039329,
@@ -339716,7 +339722,7 @@ A339804 ,0,1,4,13,22,50,68,116,162,236,278,437,498,634,794,1018,1118,1450,1574,1
A339805 ,5,17,47,97,98,159,279,359,485,489,749,879,1679,1979,2399,2499,3968,5669,6749,7199,7799,8099,8639,9719,12799,19199,25599,31999,37499,39599,44799,68599,78399,78749,79379,94499,134999,143999,146999,161999,172799,175999,194399,199679,209999,218699,259999,
A339806 ,1,2,3,7,43,239,1663,9242,47523,351115,2015403,4026914,10143015,72872619,144151023,413384223,
A339807 ,1,2,11,5,10,154,540,581,272,49,122,3418,27304,90277,150948,150519,95088,37797,8714,893,3346,142760,1938178,12186976,42696630,94605036,145009210,161845163,134933733,84656743,39632149,13481441,3156845,455917,30649,
-A339808 ,1,2,3,6,10,18,34,55,104,176,320,592,1071,1855,3311,5943,
+A339808 ,1,2,3,6,10,18,34,55,104,176,320,592,1071,1855,3311,5943,10231,
A339809 ,0,1,2,5,4,9,14,29,6,13,20,41,34,69,104,209,10,21,32,65,54,109,164,329,76,153,230,461,384,769,1154,2309,12,25,38,77,64,129,194,389,90,181,272,545,454,909,1364,2729,142,285,428,857,714,1429,2144,4289,1000,2001,3002,6005,5004,10009,15014,30029,16,33,50,
A339810 ,1,2,2,4,4,6,2,6,2,12,2,6,6,24,6,6,6,32,6,24,2,12,6,12,12,30,2,384,2,6,2,12,4,6,6,64,6,6,2,60,2,48,6,6,12,60,2,6,30,12,2,210,2,96,2,216,30,30,6,180,2,6,2,16,6,12,2,60,4,6,2,6,6,12,6,120,6,24,6,30,2,240,6,6,30,12,6,60,2,30,2,48,
A339811 ,1,2,2,3,3,4,2,4,2,5,2,4,4,6,4,4,4,7,4,6,2,5,4,5,5,8,2,9,2,4,2,5,3,4,4,10,4,4,2,11,2,12,4,4,5,11,2,4,8,5,2,13,2,14,2,15,8,8,4,16,2,4,2,17,4,5,2,11,3,4,2,4,4,5,4,18,4,6,4,8,2,19,4,4,8,5,4,11,2,8,2,12,4,4,4,11,20,4,5,19,3,4,4,4,4,
@@ -339793,7 +339799,7 @@ A339883 ,24,72,25440,33840,38880,48960,99360,123120,208320,458640,510720,519360,
A339884 ,1,1,1,1,1,1,0,2,1,1,0,1,2,1,1,0,1,2,2,1,1,0,0,2,2,2,1,1,0,0,1,3,2,2,1,1,0,0,1,2,3,2,2,1,1,0,0,0,2,3,3,2,2,1,1,0,0,0,1,3,3,3,2,2,1,1,0,0,0,1,2,4,3,3,2,2,1,1,
A339885 ,1,1,1,0,1,1,0,1,1,1,-1,0,1,1,1,0,-1,1,1,1,1,-1,-1,-1,1,1,1,1,0,-1,-1,0,1,1,1,1,0,-1,-2,-1,0,1,1,1,1,0,1,-1,-2,0,0,1,1,1,1,0,0,0,-2,-2,0,0,1,1,1,1,
A339887 ,1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,2,2,3,1,2,2,2,1,4,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,5,1,2,2,1,2,4,1,2,2,4,1,3,1,2,2,2,2,4,1,2,1,2,1,5,2,2,2,
-A339888 ,1,1,3,5,13,23,55,104,236,470,1039,
+A339888 ,1,1,3,5,13,23,55,104,236,470,1039,2140,4712,9962,21961,47484,105464,232324,521338,1167825,2651453,6031136,13863054,31987058,74448415,174109134,410265423,971839195,2317827540,5558092098,13412360692,32542049038,79424450486,
A339889 ,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,
A339890 ,0,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,1,2,2,1,2,1,4,1,1,1,4,1,1,1,3,1,2,1,2,2,1,1,6,1,2,1,2,1,3,1,3,1,1,1,5,1,1,2,5,1,2,1,2,1,2,1,8,1,1,2,2,1,2,1,6,2,1,1,5,1,1,1,
A339891 ,1,4,7,12,20,34,74,131,260,524,1030,2054,4118,8196,16389,32804,65554,131074,262216,524292,1048580,2097304,4194312,8388619,16777478,33554436,67108906,134218244,268435464,536870914,1073742880,2147483720,4294967300,8589936646,17179869193,
@@ -339855,7 +339861,7 @@ A339946 ,2,24,812,52920,5635002,889789866,195289709624,56872979140536,2122230852
A339947 ,1,5,5,13,5,33,23,30,25,69,23,150,79,119,161,385,125,501,178,443,548,1105,273,1119,921,1339,1202,2049,228,2237,2041,2792,2431,3096,1006,5905,4216,5230,3433,7596,1531,10026,6556,6939,8201,14190,3105,13431,7068,12673,12587,22075,4080,17211,13183,19462,18667,29950,2709,34199,
A339948 ,1,1,4,7,4,16,4,16,10,16,4,40,4,16,16,36,4,40,4,40,16,16,4,80,10,16,20,40,4,64,4,52,16,16,16,
A339949 ,2,3,5,6,7,3,2,12,4,4,4,4,18,2,3,6,20,5,3,2,30,4,3,4,4,9,2,3,9,4,4,3,4,47,2,3,5,10,6,3,2,15,4,4,4,4,13,2,3,7,8,5,3,2,77,4,3,5,6,8,3,2,10,4,4,3,4,24,2,3,6,78,6,3,2,22,4,3,4,4,11,2,
-A339950 ,1,7,14,20,27,35,41,48,54,62,69,75,82,90,96,
+A339950 ,1,7,14,20,27,35,41,48,54,62,69,75,82,90,96,103,109,117,124,130,137,143,151,158,164,171,179,185,192,198,206,213,219,226,234,240,247,253,260,268,274,281,287,295,302,308,315,323,329,336,342,350,357,363,370,376,384,391,397,404,
A339951 ,1,2,4,8,64,70,280,287,0,2,16,16,32,40,280,280,560,561,3366,3367,20202,20205,40410,40414,0,2,16,16,32,40,0,5,30,30,150,156,156,159,477,483,2898,2901,8703,8709,60963,60965,0,2,0,2,4,4,8,8,40,44,0,4,4,4,
A339952 ,5,13,17,25,29,37,41,45,53,61,65,73,85,89,97,101,109,113,117,125,137,145,149,153,157,169,173,181,185,193,197,205,221,225,229,233,241,245,257,261,265,269,277,281,289,293,305,313,317,325,333,337,349,353,365,369,373,377,
A339953 ,5,7,11,13,29,47,67,71,73,83,131,151,233,307,461,467,479,571,577,587,613,619,643,727,853,947,953,967,991,1063,1093,1231,1249,1291,1297,1427,1489,1493,1867,1871,1879,2017,2083,2111,2251,2309,2311,2473,2749,2753,2767,3011,3089,3191,3313,3691,
@@ -340349,6 +340355,7 @@ A340454 ,1,0,1,1,1,-1,2,0,1,1,1,0,1,-1,1,2,1,0,2,-1,1,0,1,0,2,0,0,2,2,-1,1,0,1,0
A340455 ,1,-1,2,0,0,0,2,-2,2,1,0,0,1,-2,2,0,2,0,2,-2,0,0,0,2,2,-2,2,0,-1,0,4,-2,2,-1,0,0,0,0,2,0,2,0,2,-2,2,0,-2,0,2,-2,2,2,0,0,2,-2,2,1,2,-2,0,-2,2,0,1,2,2,-2,0,0,0,0,2,-2,4,
A340456 ,1,2,2,1,2,2,2,1,2,2,2,0,2,4,2,0,1,2,2,2,2,2,2,0,3,2,2,0,2,2,2,2,2,2,0,2,2,4,2,-1,2,2,2,0,2,2,4,0,2,4,2,1,0,0,2,2,2,4,2,0,2,2,2,0,2,2,2,2,4,2,0,0,1,4,2,0,2,2,2,2,2,0,2,2,2,4,
A340457 ,0,0,1,4356,164025,
+A340458 ,1,2,2,3,4,3,4,5,5,4,5,6,6,7,5,6,7,7,8,8,6,7,8,8,9,9,9,7,8,9,9,10,10,10,11,8,9,10,10,11,11,11,12,12,9,10,11,11,12,12,12,13,13,14,10,11,12,12,13,13,13,14,14,15,14,11,12,13,13,14,14,14,15,15,16,15,16,12,13,14,14,15,15,15,16,16,17,16,17,17,
A340459 ,3,9,10,18,26,23,31,44,50,40,48,68,74,80,61,69,98,104,110,116,86,94,134,140,146,152,158,115,123,176,182,188,194,200,206,148,156,224,230,236,242,248,254,260,185,193,278,284,290,296,302,308,314,320,226,234,338,
A340460 ,1,14,36,58,80,168,190,212,234,256,278,300,322,344,611,633,655,677,699,988,1010,1032,1054,1365,1387,1409,1720,1742,1764,2075,2097,2119,2452,2474,2807,2829,3162,3184,3517,3539,3872,3894,4227,4249,4582,4604,4937,4959,5292,5314,5647,5669,6002,
A340461 ,1,0,3,2,9,0,17,6,15,4,25,2,43,10,15,14,45,6,59,10,35,14,49,6,59,30,51,28,83,0,113,30,51,28,85,20,145,40,81,22,139,14,149,40,75,26,97,14,143,34,75,68,143,24,125,64,125,54,121,2,275,82,119,62,183,18,221,58,99,
@@ -340593,7 +340600,7 @@ A340704 ,3,6,3,5,3,4,7,4,10,4,4,3,3,4,3,3,4,4,4,4,4,4,4,4,4,4,3,3,4,4,3,3,3,4,4,
A340705 ,5,4,7,8,10,3,3,3,3,3,7,4,4,3,5,4,3,3,7,3,3,5,3,3,3,3,4,4,3,3,4,4,4,3,3,4,3,4,4,3,3,4,4,3,3,4,4,4,3,3,3,4,3,3,3,3,4,4,3,4,4,4,5,3,4,4,3,4,4,4,4,4,5,4,5,4,3,4,4,3,3,4,4,4,4,3,3,3,3,4,
A340706 ,5,17,3,13,24,35,10,28,270,631,95,443,531,440,1487,1934,503,8138,6276,12311,16911,33892,11573,17000,3807,45197,30753,31457,65170,105597,127209,206808,109516,139456,377711,530040,561600,690742,952332,457704,671064,353107,
A340707 ,0,1,-1,2,0,1,-2,3,2,-2,0,8,12,-8,-7,14,-1,10,2,4,6,-3,20,-2,5,-5,-27,4,-16,5,5,4,-8,11,13,-8,-19,8,-36,3,2,-14,-5,2,-3,-55,-19,-6,14,-54,-13,-53,63,-26,38,-2,21,38,-30,7,39,2,-23,41,2,-8,5,5,-5,-110,
-A340708 ,1,2,3,5,8,13,24,40,69,130,231,408,689,1272,2153,3960,6993,
+A340708 ,1,2,3,5,8,13,24,40,69,130,231,408,689,1272,2153,3960,6993,12560,
A340709 ,0,1,2,3,5,4,7,6,10,8,12,9,15,11,17,13,20,14,22,16,25,18,27,19,30,21,32,23,35,24,37,26,40,28,42,29,45,31,47,33,50,34,52,36,55,38,57,39,60,41,62,43,65,44,67,46,70,48,72,49,75,51,77,53,80,54,82,56,85,58,87,
A340710 ,1,7,5,5,1,7,3,8,4,1,1,6,8,7,3,7,7,7,6,6,0,7,4,7,2,1,2,2,8,4,0,5,2,3,7,0,1,1,1,5,1,1,8,1,3,9,4,5,5,4,3,9,9,1,5,5,8,1,7,9,0,6,2,1,6,1,7,5,6,8,6,2,1,6,4,6,4,5,1,1,9,2,7,5,9,7,9,9,0,2,4,8,5,2,5,6,3,9,7,6,9,6,3,6,8,9,5,1,6,8,2,5,3,0,2,5,1,5,1,1,
A340711 ,1,2,7,3,9,8,6,6,1,3,2,0,6,8,3,3,9,2,5,1,5,8,1,6,8,3,8,2,1,3,8,9,4,7,2,7,3,4,7,6,2,7,4,4,4,6,7,6,7,3,5,7,8,9,4,0,0,2,9,6,8,1,4,4,0,9,8,7,4,8,6,6,8,1,5,3,7,7,6,0,6,9,5,5,6,2,0,1,2,2,8,5,4,3,8,1,1,4,6,6,0,7,3,0,5,9,2,7,4,0,5,9,2,2,4,4,6,8,1,3,
@@ -340611,7 +340618,7 @@ A340722 ,1,1,6,4,2,2,9,7,1,3,7,2,5,3,0,3,3,7,3,6,3,6,3,2,0,9,3,8,2,6,8,4,5,8,6,9
A340723 ,2,9,9,1,5,6,8,9,8,7,6,8,7,5,9,0,6,2,8,3,1,2,5,1,6,5,1,5,9,0,4,9,1,7,7,9,1,1,1,2,8,0,6,0,2,4,9,2,1,7,1,5,1,1,2,7,4,4,1,1,9,6,5,0,9,5,6,3,8,8,7,6,7,8,7,6,3,2,0,2,1,7,9,
A340724 ,1,2,9,8,0,5,5,3,3,2,6,4,7,5,5,7,7,8,5,6,8,1,1,7,1,1,7,9,1,5,2,8,1,1,6,1,7,7,8,4,1,4,1,1,7,0,5,5,3,9,4,6,2,4,7,9,2,1,6,4,5,3,8,8,2,5,4,1,6,8,1,5,0,8,1,8,9,7,5,7,9,8,6,
A340725 ,1,0,6,8,6,2,8,7,0,2,1,1,9,3,1,9,3,5,4,8,9,7,3,0,5,3,3,5,6,9,4,4,8,0,7,7,8,1,6,9,8,3,8,7,8,5,0,6,0,9,7,3,1,7,9,0,4,9,3,7,0,6,8,3,9,8,1,5,7,2,1,7,7,0,2,5,4,4,7,5,6,6,9,1,
-A340726 ,1,2,6,15,42,143,399,1190,4209,13130,41591,118590,404471,1158696,3893831,12222320,39428991,
+A340726 ,1,2,6,15,42,143,399,1190,4209,13130,41591,118590,404471,1158696,3893831,12222320,39428991,123471920,397952081,1297210320,
A340727 ,12,48,240,1440,8640,60480,604800,5443200,59875200,718502400,9340531200,124540416000,1743565824000,29640619008000,502146957312000,8536498274304000,162193467211776000,3406062811447296000,68121256228945920000,1498667637036810240000,
A340728 ,0,0,1,1,0,1,0,2,0,1,0,1,0,2,1,0,0,3,0,1,0,0,0,3,0,1,0,1,0,3,0,1,0,0,1,1,0,2,0,1,0,3,0,2,0,0,0,3,0,2,0,0,0,2,0,0,0,0,0,4,0,2,1,0,0,3,0,2,0,1,0,1,0,1,0,0,0,3,0,3,0,0,0,3,0,1,0,1,0,3,0,1,0,0,0,1,0,3,1,
A340729 ,1,3,8,18,60,150,210,420,390,840,7770,5460,9282,2310,3570,2730,10710,39270,117810,60060,154770,43890,53130,46410,66990,62790,176358,106260,30030,642180,1111110,1919190,930930,1688610,1360590,1531530,1291290,570570,1138830,510510,690690,1141140,870870,
@@ -340633,6 +340640,7 @@ A340744 ,31,23,19,43,73,53,43,37,61,43,83,73,43,73,53,67,79,73,61,59,173,151,109
A340745 ,0,2,3,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,
A340746 ,24,40,60,67,88,100,132,136,147,150,184,204,220,227,232,276,307,323,328,330,340,376,387,424,460,472,492,499,510,547,550,564,568,580,627,636,664,675,690,707,708,712,726,748,767,
A340747 ,24,40,60,67,72,88,96,100,120,132,136,144,147,150,160,168,180,184,200,204,216,220,227,232,240,264,267,276,280,288,300,307,312,323,328,330,340,348,352,360,367,376,384,387,395,396,400,408,420,424,
+A340748 ,2,3,4,22,94,142,262,334,358,694,862,934,1174,1678,1822,2182,2854,3022,3862,3958,4054,4702,4894,5062,5398,5854,6022,6238,6382,6694,7534,7558,7822,8038,8422,9502,9934,10078,10342,10558,11062,11758,12574,12622,13942,14038,
A340749 ,6,10,12,14,15,18,20,21,22,26,28,30,33,34,36,38,39,42,45,46,48,50,51,52,54,55,56,57,58,62,65,66,68,69,70,72,74,75,78,80,82,84,85,86,87,90,91,93,94,95,98,100,102,105,106,110,111,112,114,116,118,120,122,
A340750 ,5,7,11,29,43,53,89,109,113,127,131,173,179,181,199,263,311,379,419,433,443,449,461,467,479,523,571,577,593,601,613,631,653,709,719,733,739,757,811,823,829,853,929,937,967,971,1019,1031,1049,1153,1181,1321,1381,1399,1409,1439,1451,1453,1459,
A340751 ,0,0,0,0,1,1,1,2,2,3,4,5,6,8,9,11,15,13,20,21,26,28,34,32,46,44,56,54,70,62,87,80,100,94,126,105,152,127,167,154,197,171,232,200,256,237,292,251,349,291,378,336,419,355,497,405,528,458,583,487,680,549,700,620,786,
@@ -340769,6 +340777,7 @@ A340886 ,1,1,6,76,1720,60816,3096384,214579296,19422473088,2224980891904,3146755
A340887 ,1,1,7,99,2511,99531,5680125,441226521,44766049599,5748319130283,911271895816077,174799606363478361,39903413238125862309,10690643656077551475921,3321750648705212259711063,1184831658624977151885176859,480843465699932167142334581919,
A340888 ,1,1,8,124,3456,150656,9453056,807373568,90066059264,12716049596416,2216452086693888,467465806422867968,117332539562036035584,34562989958399757647872,11807922834511544081973248,4630865359842075866336067584,2066370767828213666946077425664,
A340889 ,7,23,113,139,199,211,293,317,523,691,887,1039,1069,1129,1259,1327,1381,1637,1669,1759,1831,1933,1951,2113,2179,2297,2311,2423,2477,2503,2557,2593,2633,2861,2971,3089,3137,3229,3271,3433,3739,3889,3947,3967,4159,4177,4297,4463,4523,4733,4759,4831,
+A340890 ,1,8,5184,1719926784,990677827584000000,2495937495082991616000000000000,58001506007267709490243656115814400000000000000,23264754073069200132851692722771970253637181903994880000000000000000,
A340891 ,1,1,1,2,6,20,70,255,961,3726,14797,59986,247606,1038632,4420837,19071954,83321966,368400431,1647706426,7452622503,34082926816,157595263361,736806253045,3483636843142,16660303710511,80618576499123,394863246977469,1958369414771028,
A340892 ,1,1,2,6,22,90,394,1807,8577,41810,208218,1055418,5429926,28294906,149091449,793344134,4258741610,23043290306,125589061313,689061968319,3804200404388,21125338986694,117963378773322,662200103423786,3736364727815999,21186955753874840,
A340893 ,1,1,1,3,12,51,229,1079,5288,26768,139255,741804,4035428,22374787,126262588,724423620,4222889705,24999907277,150274982778,917156371139,5684147494421,35782117189675,228878225147773,1488242327844714,9842110656790201,
@@ -340923,6 +340932,7 @@ A341042 ,1,1,3,1,5,3,7,1,6,5,11,3,13,7,15,1,17,6,19,5,21,11,23,3,10,13,9,7,29,15
A341043 ,1,35,189,559,1241,2331,3925,6119,9009,12691,17261,22815,29449,37259,46341,56791,68705,82179,97309,114191,132921,153595,176309,201159,228241,257651,289485,323839,360809,400491,442981,488375,536769,
A341044 ,8,9,10,14,15,21,22,25,26,33,34,39,42,46,57,62,65,66,69,72,74,76,80,82,87,91,92,93,94,95,100,106,111,116,119,121,122,129,133,134,145,146,159,162,166,172,176,177,183,184,190,194,202,203,206,208,213,214,215,219,232,236,237,238,240,243,
A341045 ,1,4,6,28,45,120,496,672,6048,8128,14421,30240,32760,523776,2178540,23569920,26409026,29270772,30685402,33550336,45532800,
+A341046 ,1,36,106,29486,32876,66317,1360120,22060516,78256779,1151791169,6701487259,6701487259,1142027682075,2851718461558,91822653867264,136308121570117,1952799169684491,21208174623389167,842468587426513207,842468587426513207,84383735478118508040,
A341047 ,3,113,333,92633,103283,208341,4272943,69305155,245850922,3618458675,21053343141,21053343141,3587785776203,8958937768937,288469374822515,428224593349304,6134899525417045,66627445592888887,2646693125139304345,2646693125139304345,
A341048 ,224,2464,2912,3159,3808,4256,5152,6318,6496,8288,9184,9632,
A341050 ,1,1,1,3,1,1,3,1,5,8,1,1,3,1,5,8,1,7,21,19,1,1,3,1,5,8,1,7,21,20,1,9,40,81,43,1,1,3,1,5,8,1,7,21,20,1,9,40,81,47,1,11,65,208,295,94,1,1,3,1,5,8,1,7,21,20,1,9,40,81,48,1,11,65,208,297,107,1,13,96,425,1024,1037,201,
@@ -341074,6 +341084,9 @@ A341208 ,9,12,43,101,276,711,1873,4892,12819,33549,87844,229967,602073,1576236,4
A341209 ,1,5,12,23,39,61,90,127,173,229,296,375,467,573,694,831,985,1157,1348,1559,1791,2045,2322,2623,2949,3301,3680,4087,4523,4989,5486,6015,6577,7173,7804,8471,9175,9917,10698,11519,12381,13285,14232,15223,16259,17341,18470,19647,
A341210 ,3,29,41,73,113,157,167,173,199,599,607,617,1213,1747,1979,2027,2237,2377,2441,2593,2659,2689,2693,3061,3137,3413,3457,3539,3673,3733,3769,4091,4157,4273,4289,4547,4603,4759,4877,4909,4957,5039,5231,5233,5303,5419,
A341211 ,3,3,3,13,3,3,3,113,331,3631,827,3109,4253,7487,71,
+A341212 ,154379,1075198,4211518,4700758,4745227,5954379,6036043,6330235,6485998,6524878,6851227,7846798,8536027,8556358,11718598,12100027,12126838,13584838,14869379,15320587,16934998,17074379,18154379,18904027,19013129,19774379,19779995,
+A341213 ,1,7,47,1019,154379,59423129,3100501318,126544656838,
+A341214 ,2,7,47,1019,55414379,
A341215 ,5,7,11,19,29,31,37,43,53,113,127,163,173,199,257,271,317,353,397,439,457,461,557,599,659,757,809,991,997,1019,1069,1129,1289,1327,1439,1447,1549,1621,1733,1747,1759,1831,1913,2027,2113,2141,2153,2309,2339,2357,2383,2423,2473,2663,2741,2801,
A341216 ,1,1,2,1,1,2,1,2,3,4,1,1,1,1,2,1,2,3,4,5,6,1,1,1,1,1,1,2,1,2,2,3,3,4,5,6,1,1,2,3,3,3,4,5,6,1,2,3,4,5,6,7,8,9,10,1,1,1,1,1,1,1,1,1,1,2,1,2,3,4,5,6,7,8,9,10,11,12,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,
A341217 ,3,11,5,1720949,22362444257,57703877539769,
@@ -341266,12 +341279,15 @@ A341410 ,0,0,1,2,0,0,3,2,1,0,10,6,10,2,10,2,10,12,10,14,16,8,10,18,0,4,1,18,10,0
A341411 ,0,1,0,3,0,1,1,3,0,5,4,7,2,13,0,3,15,13,15,19,15,11,15,19,0,3,0,27,15,25,15,3,27,21,15,31,15,17,30,19,15,19,15,11,0,9,15,19,1,25,21,43,15,31,25,27,54,55,15,19,15,55,36,3,5,55,15,27,18,55,15,67,15,55,
A341412 ,0,1,0,3,1,1,0,3,0,1,10,7,8,7,6,3,4,13,2,15,0,3,21,19,1,13,0,7,21,1,21,3,12,23,21,31,21,15,12,35,21,13,21,31,36,45,21,19,0,1,33,39,21,31,46,35,42,33,21,55,21,29,0,3,46,49,21,31,27,21,21,67,21,17,
A341413 ,0,0,1,0,3,2,0,4,1,0,6,8,2,0,4,4,11,14,9,16,7,8,5,20,8,10,1,0,28,20,28,4,25,4,14,32,28,26,4,36,28,20,28,12,28,2,28,20,0,0,19,48,28,32,34,28,43,24,28,56,28,16,28,4,18,20,28,52,25,0,28,68,28,66,19,40,
+A341414 ,0,1,3,8,1,5,4,7,7,4,5,1,8,3,1,0,9,7,2,9,5,6,3,3,6,5,9,2,7,9,0,1,3,8,1,5,4,7,7,4,5,1,8,3,1,0,9,7,2,9,5,6,3,3,6,5,9,2,7,9,0,1,3,8,1,5,4,7,7,4,5,1,8,3,1,0,9,7,2,9,5,6,3,3,6,5,9,2,7,9,
A341415 ,1,0,2,2,0,4,4,8,0,8,14,16,24,0,16,44,64,48,64,0,32,148,208,216,128,160,0,64,504,736,720,640,320,384,0,128,1750,2592,2672,2176,1760,768,896,0,256,6156,9280,9696,8448,6080,4608,1792,2048,0,512,
A341416 ,1,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,35,36,47,49,40,59,61,52,45,71,56,79,55,68,89,63,65,103,107,92,77,121,72,127,85,91,137,139,88,151,112,124,115,169,104,119,99,148,193,197,133,211,223,117,145,161,136,241,155,196,
A341417 ,1,2,2,3,2,3,1,2,0,2,0,4,2,7,5,10,6,11,5,10,3,10,3,13,6,19,11,25,14,29,14,30,12,31,12,36,16,45,23,56,30,65,33,71,34,76,35,84,40,97,48,113,58,129,66,144,71,157,76,172,83,192,
A341418 ,1,1,1,0,2,1,0,1,3,1,-1,0,3,4,1,0,-2,1,6,5,1,-1,-2,-3,4,10,6,1,0,-2,-6,-3,10,15,7,1,0,-2,-6,-12,0,20,21,8,1,0,1,-6,-16,-19,9,35,28,9,1,0,0,0,-16,-35,-24,28,56,36,10,1,1,2,3,-6,-40,-65,-21,62,84,45,11,
A341419 ,1,1,2,0,4,2,0,-2,8,6,8,-2,0,-2,-8,-2,16,14,24,-2,32,14,-8,-18,0,-2,-8,-2,-32,-18,-8,14,32,30,56,-2,96,46,-8,-50,128,94,120,-34,-32,-50,-136,-18,0,-2,-8,-2,-32,-18,-8,14,-128,-98,-136,30,-32,14,120,46,64,62,
A341420 ,1,4,5,8,13,17,20,25,29,37,40,41,52,53,61,65,68,73,85,89,97,100,101,104,109,113,116,125,136,137,145,148,149,157,164,169,173,181,185,193,197,200,205,212,221,229,232,233,241,244,257,260,265,269,277,281,289,292,293,296,
+A341421 ,0,0,0,-1,-2,-2,-2,-3,-3,-2,-1,-2,-2,-3,-2,-2,-2,-2,-2,-2,-2,-3,-3,-3,-3,-4,-4,-5,-5,-5,-6,-5,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,-1,-1,-1,-1,-1,-2,-3,-3,-3,-3,-3,-3,-4,-5,-4,-5,-6,-7,
+A341422 ,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,2,2,2,2,2,4,2,2,2,2,2,4,4,2,4,2,2,2,4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,2,2,2,4,2,2,2,2,2,4,2,4,2,2,2,2,4,2,2,2,2,4,4,2,2,2,4,2,2,
A341423 ,1,5,32,94,219,437,804,1362,2177,3271,4768,6708,9227,12381,16254,20954,26707,33461,41480,50884,61703,74183,88606,104862,123481,144241,167604,193648,222799,254731,290244,329512,372545,419661,470822,526646,587481,653505,
A341424 ,6,51,177,547,1348,2958,5574,10084,16974,27450,41970,62671,90216,128082,175867,238018,316373,414998,534094,682144,859705,1075165,1326551,1627896,1976582,2390057,2862607,3411273,4039483,4760419,5571729,6500650,7541560,8722096,
A341425 ,7,48,331,1269,3698,9382,20927,42683,79844,142173,238810,387615,603589,915324,1345294,1939221,2729723,3783313,5138567,6895632,9108626,11909496,15362753,19642539,24832744,31179476,38757032,47877886,58647957,71447776,86391220,
@@ -341299,6 +341315,7 @@ A341446 ,2,5,6,11,14,17,18,23,26,31,35,38,41,42,47,54,58,59,65,67,73,74,78,83,86
A341447 ,3,7,13,15,19,29,33,37,43,51,53,61,69,71,75,77,79,89,93,101,107,113,119,123,131,139,141,151,161,163,165,173,177,181,193,199,201,217,219,221,223,229,239,249,251,255,263,271,281,287,291,293,299,309,311,317,
A341448 ,6,14,15,24,26,33,35,38,51,54,56,58,60,65,69,74,77,86,93,95,96,104,106,119,122,123,126,132,135,140,141,142,143,145,150,152,158,161,177,178,185,201,202,204,209,214,215,216,217,219,221,224,226,232,234,240,
A341449 ,1,5,11,17,23,25,31,41,47,55,59,67,73,83,85,97,103,109,115,121,125,127,137,149,155,157,167,179,187,191,197,205,211,227,233,235,241,253,257,269,275,277,283,289,295,307,313,331,335,341,347,353,365,367,379,389,
+A341450 ,1,0,0,0,0,1,0,2,1,3,3,6,3,9,9,12,12,20,18,28,27,37,42,55,51,74,80,98,105,136,137,180,189,232,255,308,320,403,434,512,551,668,706,852,915,1067,1170,1370,1453,1722,1860,2145,2332,2701,2899,3355,3626,4144,
A341451 ,1,0,0,1,0,1,1,1,2,2,2,3,3,4,6,5,7,8,9,10,13,13,17,17,22,21,27,27,34,34,41,40,51,49,62,59,71,70,86,82,101,97,117,112,135,131,155,150,180,170,202,196,228,222,259,248,291,281,324,314,361,348,404,388,445,431,
A341452 ,1,0,0,1,0,1,1,1,2,2,2,3,3,4,6,6,7,9,9,12,14,16,18,22,24,29,31,38,40,49,50,62,65,77,81,97,98,120,122,144,149,176,178,212,214,251,255,299,304,352,355,412,417,482,485,559,564,643,650,742,745,850,856,965,
A341453 ,1,0,0,1,0,1,1,1,2,2,2,3,3,4,6,6,7,9,10,12,15,16,20,23,27,30,36,40,48,53,62,68,81,87,105,112,130,141,166,176,208,219,256,271,314,331,385,403,468,488,561,588,674,702,804,837,952,991,1126,1168,1321,1372,
@@ -341401,6 +341418,7 @@ A341549 ,3,6,23,50,131,294,687,1530,3419,7502,16391,35490,76467,163830,349535,74
A341550 ,29,103,1229,2609,3733,4229,4903,11239,21013,47507,65033,73453,75629,105601,112241,132499,172213,257069,330641,361213,379459,570029,667477,893033,950633,976147,1054717,1240999,1435219,1934837,2149151,2775559,2829011,3189799,
A341551 ,996787,87880249,6458329435,437811072433,28577902283587,1831839463314409,116388761878654315,7363089071153371873,464825043098493809107,29313469954934882953369,1847663299656911486659195,116431149842916469716759313,7336041758469840870854326627,
A341552 ,129,975,7041,49935,351489,2466255,17281281,121021455,847307649,5931625935,41522798721,290663842575,2034659652609,14242655832015,99698705615361,697891283681295,4885240018890369,34196683231596495,239376791919267201,1675637571329145615,
+A341553 ,3451,61567,996787,15478951,235916971,3565011727,53659360867,806180862391,12101749545691,181589509846687,2724285545507347,40867383560793031,613032456339776011,9195638766433606447,137935644948388268227,2069042118396589446871,
A341554 ,1,6,-810,-22134,-278634,
A341555 ,0,1,-36,-81,784,-1314,
A341556 ,1,72,2376,47592,646344,
@@ -341432,6 +341450,7 @@ A341581 ,0,1,2,5,10,20,37,70,130,243,450,836,1549,2874,5326,9875,18302,33928,628
A341582 ,0,1,2,4,6,12,22,42,76,142,262,488,902,1674,3100,5750,10654,19752,36606,67858,125772,233134,432118,800968,1484630,2751866,5100732,9454534,17524526,32482792,60208782,111600642,206858476,383424702,710700742,1317326728,2441744422,
A341583 ,0,1,3,8,18,42,94,208,450,966,2052,4330,9074,18920,39266,81182,167268,343634,704122,1439496,2936906,5981174,12161332,24691514,50066690,101400616,205150098,414653998,837377988,1689714242,3407154474,6865700808,13826659450,27829885126,
A341584 ,0,1,2,2,2,2,4,3,4,4,4,4,4,4,6,4,6,5,8,6,8,6,8,6,8,7,8,8,8,8,8,8,9,8,10,9,10,10,12,10,11,9,12,9,12,10,12,10,13,10,14,11,14,11,16,11,16,12,16,12,16,13,16,13,16,14,16,13,16,14,18,14,16,14,20,14,16,14,20,15,18,
+A341585 ,1,0,1,1,0,2,3,0,4,4,0,5,
A341586 ,1,0,-4,-5,22,98,-5,-1458,-5136,9053,161328,549822,-1954067,-30099188,-114161728,500200027,8875931202,42311243830,-149028931789,-3816065804086,-24704581255020,33033659868037,2184285021783940,20047242475274290,30117550563701293,
A341587 ,1,6,40,315,2908,30989,375611,5112570,77305024,1286640410,23387713930,461187042992,9808283703684,223833267479764,5456669750439788,141540592345674800,3892707724320135616,113153294901088030320,3466501398608272647984,111636571036702743967104,3770483138507706753943584,
A341588 ,1,12,130,1485,18508,253400,3805723,62437500,1113510409,21479997957,446094038806,9930796412082,236037249893092,5968192832899412,160007282538148508,4534905316824903144,135500246340709682692,4257646241716404353684,140366073694357927723936,4845119946789226304526392,
@@ -341479,6 +341498,7 @@ A341629 ,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1
A341630 ,2,3,6,14,36,94,250,675,1832,5005,13746,37901,104902,291312,811346,2265905,6343854,17801383,50057400,141034248,398070362,1125426581,3186725646,9036406687,25658313188,72946289247,207628101578,591622990214,1687527542874,4818113792640,
A341631 ,2,7,9,14,19,27,28,29,30,36,44,60,61,68,70,71,87,88,89,100,101,104,105,108,109,112,113,138,157,174,192,193,199,201,202,203,204,210,274,275,276,277,304,305,306,364,365,366,372,373,384,387,388,389,399,400,401,405,471,472,473,511,512,513,
A341632 ,5,31,41,59,139,281,281,281,311,479,593,1153,1153,1283,1559,1559,2213,2213,2213,3167,3167,2963,2963,3067,3067,3181,3181,5153,6779,7451,9463,9463,9859,10061,10061,10061,10061,10889,17539,17539,17539,17539,22229,22229,22229,30869,30869,30869,32717,
+A341633 ,1,2,4,24,621,492288,81203064840,
A341634 ,101,11,2,3,41,5,23,7,181,19,251,43,127,53,281,29,541,37,83,11551,139,47,523,1481,157,149,12451,67,59,283,11177,2551,239,1187,1453,79,881,257,89,1553,2851,199,347,563,1483,277,14551,1753,269,827,853,15551,367,
A341635 ,1,-2,-3,1,-5,6,-7,0,2,10,-11,-3,-13,14,15,0,-17,-4,-19,-5,21,22,-23,0,4,26,0,-7,-29,-30,-31,0,33,34,35,2,-37,38,39,0,-41,-42,-43,-11,-10,46,-47,0,6,-8,51,-13,-53,0,55,0,57,58,-59,15,-61,62,-14,0,65,
A341636 ,1,4,6,13,10,24,14,38,29,40,22,78,26,56,60,103,34,116,38,130,84,88,46,228,79,104,124,182,58,240,62,264,132,136,140,377,74,152,156,380,82,336,86,286,290,184,94,618,153,316,204,338,106,496,220,532,228,232,118,780,122,248,
@@ -341539,6 +341559,7 @@ A341690 ,16,95591589000729770,57770815231373815452404527382911050,15942241394469
A341691 ,0,1,2,1,4,1,2,5,8,1,2,9,4,9,10,5,16,1,2,17,4,17,18,5,8,17,18,9,20,9,10,21,32,1,2,33,4,33,34,5,8,33,34,9,36,9,10,37,16,33,34,17,36,17,18,37,40,17,18,41,20,41,42,21,64,1,2,65,4,65,66,5,8,65,
A341692 ,1,10,100,101,102,20,103,104,105,11,106,120,2,200,107,108,30,13,109,40,14,110,50,15,12,16,130,60,61,21,201,202,22,203,70,140,150,170,17,160,80,38,230,180,31,113,190,90,49,204,210,41,114,18,301,205,250,310,
A341694 ,1,1,1,2,1,1,2,2,2,1,1,1,2,3,1,1,1,3,2,5,1,3,2,1,4,2,8,1,3,3,3,3,7,2,13,1,1,1,3,5,5,11,2,21,1,1,2,4,3,8,9,18,2,34,1,2,1,1,5,3,13,17,29,2,55,1,2,1,1,4,9,3,21,31,47,2,89,1,
+A341695 ,1,1,1,2,2,2,4,6,6,6,8,16,22,22,22,16,40,68,90,90,90,32,96,192,304,394,394,394,64,224,512,928,1412,1806,1806,1806,128,512,1312,2656,4552,6752,8558,8558,8558,256,1152,3264,7264,13712,22664,33028,41586,41586,41586,
A341696 ,2,1,4,6,40,20,46,8,42,400,60,62,64,26,406,80,48,4000,
A341697 ,1,1,1,1,2,2,4,4,6,7,11,11,17,17,25,29,38,38,54,54,72,80,102,102,136,140,174,186,228,228,300,300,366,388,464,480,594,594,702,736,874,874,1068,1068,1250,1324,1528,1528,1828,1844,2144,2220,2534,2534,2982,3026,3464,3572,4028,4028,
A341698 ,1,1,-1,1,-2,2,0,0,-2,1,3,-3,1,-1,1,-5,4,-4,12,-12,14,-14,8,-8,10,-14,12,-16,18,-18,26,-26,36,-30,22,-22,24,-24,0,2,20,-20,-10,10,12,-18,2,-2,14,-14,-2,10,16,-16,-8,20,14,10,-46,46,-52,52,-104,132,-70,74,-186,186,-134,150,
@@ -341613,6 +341634,7 @@ A341767 ,1,4,9,4,2,9,7,1,9,11,22,39,41,54,69,71,88,99,11,41,93,77,78,99,44,11,99
A341768 ,0,-2,-2,3,16,40,78,133,208,306,430,583,768,988,1246,1545,1888,2278,2718,3211,3760,4368,5038,5773,6576,7450,8398,9423,10528,11716,12990,14353,15808,17358,19006,20755,22608,24568,26638,28821,31120,33538,36078,38743,41536,44460,
A341769 ,3,12,64,436,3624,35516,400544,5106180,72574936,1137563980,19489399824,362279121044,7261032943688,156078126597084,3581487541784704,87378336982197028,2258453972652164280,61646205047945592428,1771962416919392083184,53498826047517147678132,
A341770 ,1,8,23,34,61,62,97,138,189,248,315,390,473,564,663,770,885,1008,1139,1278,1425,1580,1743,1914,2093,2280,2475,2678,2889,3108,3335,3570,3813,4064,4323,4590,4865,5148,5439,5738,6045,6360,6683,7014,7353,7700,8055,8418,
+A341771 ,2,1,1,2,2,2,3,1,2,3,2,3,2,4,1,2,4,2,3,2,3,2,4,2,5,1,2,4,2,3,3,3,3,4,2,3,3,2,4,3,3,3,3,4,3,4,2,5,2,5,2,3,4,3,5,2,6,1,2,3,3,3,4,2,3,6,2,3,3,3,4,3,5,2,4,3,7,1,2,3,4,2,3,4,2,3,3,8,
A341772 ,1,4,10,17,28,40,54,70,94,112,130,170,180,216,280,284,304,376,378,476,540,520,550,700,716,720,858,918,868,1120,990,1144,1300,1216,1512,1598,1404,1512,1800,1960,1720,2160,1890,2210,2632,2200,2254,2840,2682,2864,3040,3060,2860,3432,3640,
A341773 ,1,0,0,1,0,0,1,0,0,2,0,0,2,0,0,2,0,0,3,1,0,3,1,0,3,1,0,4,2,0,4,2,0,4,3,0,5,4,1,5,4,1,5,5,1,6,6,2,6,6,2,6,7,3,7,9,4,8,9,4,8,10,5,9,12,6,10,12,7,10,13,8,12,15,10,13,16,11,13,17,12,
A341774 ,1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,1,0,0,1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0,0,2,1,0,1,1,0,1,0,0,1,1,0,0,0,0,1,1,0,1,1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,1,0,0,2,0,0,0,0,0,2,0,0,1,1,0,2,0,0,2,
@@ -341722,6 +341744,7 @@ A341879 ,1,0,2,2,0,2,3,2,0,0,0,4,3,2,4,0,0,4,0,2,0,0,0,4,0,0,0,6,0,2,5,4,0,0,0,4
A341880 ,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,6,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,0,4,0,4,0,0,0,12,0,0,0,10,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,16,1,0,0,12,0,0,0,4,0,12,0,0,0,0,0,40,0,0,0,6,0,0,0,4,0,0,0,28,0,0,0,16,
A341881 ,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,5,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,15,
A341882 ,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,
+A341883 ,0,1,1,2,3,3,3,4,3,4,4,4,3,4,2,3,3,3,5,5,5,5,6,6,9,9,8,8,7,5,8,5,8,7,7,6,7,6,8,8,7,9,7,7,8,7,6,8,6,6,6,5,6,9,8,12,10,9,9,8,8,8,9,7,7,5,6,6,8,9,10,8,
A341884 ,1,2,6,22,342,1444,33184,399235,12502550,117906198,7740054144,74673569118,4724493959332,121637216075836,4503600768557056,89450720590507768,10119960926575526448,152232968281237988010,16384000020089600480000,552020693349464673399080,35271474934322858202723576,
A341885 ,0,3,6,6,15,9,28,9,12,18,66,12,91,31,21,12,153,15,190,21,34,69,276,15,30,94,18,34,435,24,496,15,72,156,43,18,703,193,97,24,861,37,946,72,27,279,1128,18,56,33,159,97,1431,21,81,37,196,438,1770,27,1891,499,40,18,106,75,2278,159,282,
A341886 ,512,1024,2048,5632,8192,11264,16384,22528,54272,57856,97792,108544,122368,131072,150016,165376,169984,180224,188928,195584,210432,244736,248320,256000,276480,279040,300032,317440,333312,334336,335872,337408,352256,367616,371712,
@@ -341774,6 +341797,7 @@ A341948 ,1,1,2,2,4,4,6,5,8,6,10,7,12,9,15,10,18,12,21,14,25,15,29,18,33,21,37,20
A341949 ,1,1,2,2,4,4,7,6,9,8,12,10,16,12,19,15,24,18,29,21,35,25,41,29,49,33,56,37,63,41,72,46,82,51,91,58,105,63,115,68,128,77,143,83,158,90,174,101,193,107,211,116,231,128,250,134,273,142,294,157,321,165,347,176,374,
A341950 ,1,1,2,2,4,4,7,7,10,9,14,12,19,16,23,19,30,24,37,29,44,35,55,41,65,49,75,56,89,63,102,72,116,82,134,91,153,105,171,115,194,128,220,143,242,158,273,174,305,193,334,211,374,231,412,250,447,273,494,294,541,321,
A341951 ,1,1,2,2,4,4,7,7,11,10,15,14,21,19,27,23,35,30,44,37,54,44,67,55,81,65,96,75,115,89,133,102,155,116,180,134,206,153,236,171,271,194,305,220,346,242,391,273,438,305,489,334,551,374,608,412,674,447,750,494,823,
+A341952 ,1,1,-1,1,-1,1,-1,0,1,0,-1,1,-1,0,0,1,0,-1,1,0,-1,0,1,-1,1,-1,1,-1,1,-1,0,1,-1,1,0,0,-1,0,0,1,-1,1,-1,1,0,0,-1,1,-1,0,1,-1,0,0,0,1,0,-1,1,-1,1,-1,1,-1,1,0,-1,1,-1,0,1,0,0,-1,0,1,-1,1,-1,1,-1,1,0,-1,0,1,-1,1,-1,0,1,-1,1,-1,0,0,1,0,0,-1,0,0,1,0,-1,
A341953 ,1,4,9,4,2,9,7,1,9,10,11,11,19,17,18,19,14,11,19,40,81,77,59,11,25,49,81,71,59,90,91,94,99,94,92,99,97,91,99,40,71,11,49,77,18,49,74,11,49,70,81,47,29,11,55,79,81,41,29,90,91,94,99,94,92,99,97,
A341954 ,1,2,13,99,839,7606,72190,708294,7126305,73125017,762337935,8051642336,85971106450,926481778388,10064065073450,110080177918855,1211363817278035,13401851361051323,148978925959605763,1663181275248666597,
A341955 ,1,2,11,80,659,5865,54954,534087,5334509,54423368,564713959,5941244370,63230204938,679510980507,7363532850004,80372780735971,882818219523503,9751004973855748,108236495732967482,1206750569591821120,13507907804245679450,
@@ -341811,6 +341835,7 @@ A341986 ,1,7,28,77,168,308,511,785,1155,1603,2142,2723,3430,4207,5202,6216,7497,
A341987 ,1,8,36,112,274,560,1016,1688,2647,3928,5580,7568,9990,12832,16332,20336,25167,30472,37004,44136,53054,62272,73788,85240,100276,114752,134072,151144,174834,194616,224304,247240,283467,308448,352668,381032,436368,467272,533520,
A341988 ,1,9,45,156,423,954,1887,3384,5661,8935,13446,19332,26838,36126,47691,61668,78696,98631,122665,150516,184230,222438,268146,318564,379383,445572,525942,610344,712872,817290,947166,1075680,1238148,1391475,1591236,1773684,2022241,
A341989 ,1,10,55,210,625,1542,3310,6390,11400,19090,30353,46060,67210,94780,130230,174862,230650,298800,382115,482090,603373,746860,918770,1118100,1355110,1626742,1949190,2312380,2740220,3212640,3769784,4375900,5092485,5854680,6758935,7703112,
+A341990 ,0,1,4,12,40,128,402,1278,4040,12776,40417,127803,404136,1277995,4041401,12779996,40413886,127799963,
A341991 ,1,1,1,1,2,2,1,1,4,12,6,6,6,6,3,1,8,8,12,12,6,6,
A341992 ,123,124,132,135,142,145,153,154,213,214,231,236,241,246,263,264,312,315,321,326,351,356,362,365,412,415,421,426,451,456,462,465,513,514,531,536,541,546,563,564,623,624,632,635,642,645,653,654,
A341993 ,0,1,2,4,8,3,6,5,7,9,10,20,40,80,11,22,44,88,12,24,48,96,13,26,14,28,56,15,16,17,18,19,38,76,21,42,84,168,23,46,92,184,368,25,27,29,58,30,60,31,62,32,64,128,256,33,66,34,68,35,36,37,74,148,296,39,
@@ -341841,6 +341866,7 @@ A342017 ,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2
A342018 ,8,16,24,36,44,52,64,72,80,88,92,100,108,116,120,126,128,136,144,156,164,172,184,192,200,208,216,222,224,232,244,252,260,268,271,272,280,288,296,300,308,316,324,336,344,348,352,364,372,380,388,392,397,400,408,416,424,432,440,444,448,452,460,468,476,480,488,493,496,
A342019 ,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,
A342020 ,0,2,0,1,2,6,13,32,63,124,244,453,862,1568,2835,5150,9251,16093,27830,48605,84765,145300,245730,417251,721100,1267411,2247106,3997263,
+A342021 ,5,8,41,47057,
A342022 ,1,2,2,3,4,5,2,5,6,7,8,9,4,10,11,12,13,14,15,11,16,17,18,19,20,8,21,22,23,24,2,10,25,9,21,26,27,24,28,29,30,31,18,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,4,11,8,14,49,50,21,51,52,53,54,55,56,57,58,59,60,61,7,62,63,64,65,66,
A342023 ,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,
A342024 ,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,
@@ -341898,6 +341924,7 @@ A342076 ,1,11,12,2,3,31,13,32,21,14,4,5,51,15,52,22,23,33,34,41,16,6,7,71,17,72,
A342077 ,1,2,20,3,30,4,5,50,6,7,70,8,9,10,11,12,21,13,31,14,15,51,16,17,71,18,19,22,33,40,41,42,23,32,24,43,34,44,45,52,25,53,35,54,46,47,72,26,48,49,55,60,61,62,27,73,36,63,37,74,64,65,56,66,67,75,57,76,68,69,77,80,81,82,28,83,38,
A342078 ,1,10,2,3,30,4,5,50,6,7,70,8,9,90,11,20,21,12,22,23,31,13,32,24,25,51,14,26,27,71,15,52,28,29,91,16,33,40,41,17,72,42,43,34,44,45,53,35,54,46,47,73,36,48,49,92,55,60,61,18,62,63,37,74,64,65,56,66,67,75,57,76,68,
A342079 ,1,2,21,3,4,41,5,6,61,7,8,81,9,11,12,22,23,13,14,42,24,43,15,16,62,25,17,18,82,26,63,19,31,32,27,33,34,44,45,35,36,64,46,65,37,38,83,39,51,52,28,84,47,53,54,48,85,55,56,66,67,57,58,86,68,87,59,71,72,29,73,74,49,75,76,69,77,78,88,89,
+A342080 ,18872,18874,18890,18892,22085,22111,22112,22116,22120,22121,22130,22210,22211,22220,22256,22310,22570,22571,22580,22607,22616,22652,22670,22679,22697,22706,22710,22724,22762,22825,22832,22841,22850,22859,22864,
A342081 ,1,2,3,4,5,6,7,8,10,11,13,14,16,17,19,20,22,23,26,28,29,31,32,34,37,38,41,43,44,46,47,52,53,58,59,61,62,64,67,68,71,73,74,76,79,82,83,86,88,89,92,94,97,101,103,104,106,107,109,113,116,118,122,124,
A342082 ,9,12,15,18,21,24,25,27,30,33,35,36,39,40,42,45,48,49,50,51,54,55,56,57,60,63,65,66,69,70,72,75,77,78,80,81,84,85,87,90,91,93,95,96,98,99,100,102,105,108,110,111,112,114,115,117,119,120,121,123,125,
A342083 ,1,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,3,2,2,1,4,1,2,2,3,1,4,1,3,2,2,2,4,1,2,2,4,1,5,1,3,3,2,1,6,1,3,2,3,1,5,2,4,2,2,1,7,1,2,3,3,2,5,1,3,2,4,1,8,1,2,3,3,2,5,1,6,2,2,1,7,2,2,2,
@@ -341918,6 +341945,7 @@ A342097 ,1,1,1,1,2,1,2,2,3,3,3,3,4,6,6,7,8,8,9,11,13,15,18,20,24,25,29,32,39,42,
A342098 ,1,1,1,2,2,2,3,3,3,4,5,5,6,7,7,8,9,10,11,12,13,14,15,16,18,20,21,23,25,26,28,31,33,35,38,40,42,45,48,51,55,58,61,65,68,72,77,81,85,90,94,98,104,109,114,121,127,132,139,146,
A342099 ,1,1,2,32,8704,69074944,24438162587648,546639076930132901888,1040668139730671025101058605056,218400176068773166949459169210753567686656,6353017630286823410670432558608528274164598967780769792,
A342100 ,12,18,20,24,40,42,56,60,72,80,84,88,90,102,104,108,112,114,354,366,368,372,380,384,392,396,400,402,464,468,476,480,492,500,504,552,560,564,572,576,580,582,650,654,836,840,945,948,952,954,1002,2002,2004,2024,
+A342101 ,1,2,3,1,3,1,2,1,3,1,2,3,1,1,2,1,3,1,2,3,1,3,1,2,1,1,2,3,1,1,2,1,3,1,2,3,1,3,1,2,1,3,1,2,3,1,1,2,1,1,2,3,1,3,1,2,1,1,2,3,1,1,2,1,3,1,2,3,1,3,1,2,1,3,1,2,3,1,1,2,1,3,1,2,3,1,3,
A342102 ,0,1,2,3,6,5,4,7,14,12,10,13,9,11,8,15,30,28,26,25,24,22,21,29,20,19,18,27,17,23,16,31,62,60,58,56,57,52,50,54,53,49,44,51,42,48,46,61,45,41,38,43,37,40,39,59,35,36,34,55,33,47,32,63,126,124,122,
A342103 ,1,3,6,14,15,30,35,42,56,70,78,105,140,168,190,210,248,264,270,357,418,420,570,594,616,630,714,744,812,840,910,1045,1240,1254,1485,1672,1848,2090,2214,2376,2436,2580,2730,2970,3080,3135,3339,3596,3720,3828,3956,4064,4180,
A342104 ,2,12,18630,27000,443394,6242022,14412720,22315419,26744100,44630838,50496960,106034880,128710944,148536990,162907584,212072880,218470770,296259930,349444530,397253968,535267776,641250900,641418960,666274653,684165552,688208724,709639408,
@@ -341940,6 +341968,7 @@ A342120 ,1,1,0,1,1,0,1,2,2,0,1,3,6,3,0,1,4,12,16,5,0,1,5,20,45,44,8,0,1,6,30,96,
A342121 ,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,6,0,0,9,6,0,6,4,0,0,6,0,0,0,0,0,14,0,4,13,18,0,4,0,10,5,0,17,14,0,14,12,0,8,10,0,4,0,18,12,4,0,14,0,0,0,0,0,30,0,12,21,42,0,0,33,30,1,12,21,42,0,
A342122 ,0,1,0,1,0,3,0,1,0,5,2,3,11,7,0,1,0,9,6,5,0,13,6,3,19,11,0,7,23,15,0,1,0,17,14,9,4,25,18,5,37,21,10,13,0,29,14,3,35,19,0,11,43,27,4,7,39,23,55,15,47,31,0,1,0,33,30,17,12,49,42,9,0,41,30,
A342123 ,0,0,0,0,0,0,0,0,0,0,11,0,2,0,0,0,0,0,19,0,0,9,23,0,6,4,0,0,6,0,0,0,0,0,35,0,37,13,39,0,4,0,43,5,0,17,47,0,14,12,0,8,10,0,55,0,18,12,4,0,14,0,0,0,0,0,67,0,69,21,71,0,0,33,75,1,77,21,
+A342124 ,2,18,121,124,313,484,797,2016,2211,2862,4507,6188,6325,9660,12669,13016,16857,19530,23069,28184,38761,46302,42515,49846,59087,70260,73385,78960,97267,98316,111023,124454,134641,152952,163043,180596,195975,218432,237623,
A342125 ,2,6,10,30,70,210,770,2310,10010,30030,34034,170170,510510,646646,3233230,9699690,14872858,74364290,223092870,431312882,2156564410,6469693230,13370699342,66853496710,200560490130,494715875654,2473579378270,7420738134810,20283350901814,
A342127 ,0,1,5,6,10,47,50,60,75,78,100,125,152,457,500,600,750,1000,1025,1052,1250,1520,5000,5625,6000,7500,10000,10025,10052,10250,10520,12266,12500,15200,23258,43567,50000,56250,60000,62656,75000,82291,90625,98254,100000,100025,100052,100250,100520,
A342128 ,0,1,0,2,0,0,3,2,0,0,4,6,2,0,0,5,12,18,2,0,0,6,20,84,114,2,0,0,7,30,260,2652,2970,2,0,0,8,42,630,29660,1321860,1185282,2,0,0,9,56,1302,198030,187430900,130253748108,100301050602,2,0,0,10,72,2408,932862,10199069190,2157531034816940,
@@ -341974,6 +342003,7 @@ A342159 ,0,1,4,13,40,41,172,85,464,145,980,221,1784,313,2940,421,4512,545,6564,6
A342160 ,0,7,19,26,37,56,63,98,117,124,152,189,208,215,218,279,316,335,342,387,448,485,504,511,513,604,665,702,721,728,784,875,936,973,992,999,1115,1206,1267,1304,1323,1330,1385,1512,1603,1664,1701,1720,1727,1854,1981,
A342161 ,0,1,3,4,-3,-14,63,274,-1383,-7934,50523,353794,-2702763,-22368254,199360983,1903757314,-19391512143,-209865342974,2404879675443,29088885112834,-370371188237523,-4951498053124094,69348874393137903,1015423886506852354,-15514534163557086903,
A342162 ,11,9,13,14,15,16,17,18,19,20,180,1,13,37,55,73,91,109,127,145,221,17,1,34,37,55,73,91,109,127,231,35,17,1,55,37,55,73,91,109,241,53,35,17,1,76,37,55,73,91,251,71,53,35,17,1,97,37,55,73,261,89,71,53,35,17,1,118,37,55,271,107,
+A342163 ,2,6,15,29,60,87,137,176,247,360,422,568,689,776,923,1136,1369,1494,1764,1978,2128,2451,2710,3074,3562,3870,4077,4411,4638,4995,6026,6426,6987,7271,8180,8493,9134,9802,10319,11030,11767,12139,13314,13712,14329,14742,
A342164 ,0,1,2,3,4,5,6,7,8,9,10,11,12,15,17,19,16,23,18,27,30,27,22,27,24,39,41,44,28,47,50,41,52,56,50,56,56,56,50,56,53,72,42,75,54,80,80,76,83,80,85,92,90,80,54,99,94,99,86,99,98,99,108,99,108,99,108,99,126,99,
A342165 ,1,2,3,2,4,3,5,2,4,3,6,5,7,2,4,3,8,6,9,5,7,2,10,4,3,8,6,9,11,5,12,7,2,10,4,3,13,8,6,9,14,11,15,5,12,7,16,2,10,4,3,13,17,8,6,9,14,11,18,15,19,5,12,7,16,2,20,10,4,3,21,13,22,17,8,6,9,14,
A342166 ,1,1,3,39,2925,1582425,7410496275,350464600333575,191295845123076910125,1355763582602823185129417625,138623522325287867599380791765497875,224935042709004795568466587349227029537282375,6318777956744220129890735589019782971247629409914638125,
@@ -341995,11 +342025,15 @@ A342182 ,1,1,8,117,3184,134025,8141436,672837277,72634878016,9923765772177,16738
A342183 ,3,7,11,13,31,41,43,59,73,113,139,179,197,211,223,241,263,277,349,367,449,563,587,631,659,683,739,773,823,829,977,1033,1049,1217,1471,1487,1553,1571,1583,1607,1609,1669,1697,1753,1901,1907,2089,2111,2281,2531,
A342184 ,7,13,31,97,109,157,271,523,601,691,769,829,1063,1069,1201,1249,1291,1483,1489,1567,1579,1609,1693,1747,1831,2203,2281,2383,2803,2887,2953,3511,3967,4513,4651,5023,5059,5437,5653,5779,5821,6151,6163,6199,6361,6367,
A342185 ,2,11,23,101,149,227,239,269,353,479,557,569,647,683,809,827,983,1289,1607,1619,1823,1901,1907,2039,2213,2411,2447,2843,2879,2957,2963,3011,3119,3257,3389,3557,3671,3833,3923,4001,4019,4397,4943,5099,5309,5441,5471,
+A342186 ,1,-1,1,3,-4,1,-21,31,-11,1,315,-486,196,-26,1,-9765,15381,-6562,1002,-57,1,615195,-978768,428787,-69688,4593,-120,1,-78129765,124918731,-55434717,9279163,-652999,19833,-247,1,
A342187 ,44,48,49,63,75,80,98,99,116,147,171,175,207,244,260,275,288,315,324,332,360,363,368,387,404,475,476,495,507,524,528,531,539,548,549,575,603,604,624,636,656,675,692,724,725,747,764,774,800,819,832,844,845,846,
A342188 ,80,624,2511,5264,6399,7695,7856,10287,13040,14640,15471,15632,18063,19375,20624,20816,23247,23408,25839,27135,28560,28592,31023,31184,33615,35072,36015,36368,38799,38960,39375,40816,41391,44144,46250,46575,46736,49167,51920,
A342189 ,135,296,343,351,375,512,728,999,1160,1215,1375,1431,1592,1624,2079,2240,2295,2375,2456,2624,2727,2888,2943,3104,3159,3429,3591,3624,3752,3992,4023,4184,4616,4671,4832,4887,4913,5048,5144,5319,5480,5535,5696,5831,6183,
A342190 ,1,2,3,5,7,9,10,11,12,13,14,15,17,19,23,24,26,27,29,31,34,35,37,39,40,41,43,44,46,47,49,53,55,56,58,59,61,62,63,67,68,70,71,73,74,75,76,78,79,80,81,83,89,90,94,95,97,98,100,101,103,104,107,109,110,
A342191 ,1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,21,23,24,25,27,29,30,31,32,35,36,37,41,42,43,45,47,48,49,53,54,55,59,60,61,63,64,65,67,71,72,73,75,77,79,81,83,84,89,90,91,96,97,101,103,105,107,108,109,
+A342192 ,6,10,14,22,26,34,38,46,58,62,74,82,86,94,100,106,118,122,134,140,142,146,158,166,178,194,196,202,206,214,218,220,226,254,260,262,274,278,298,300,302,308,314,326,334,340,346,358,362,364,380,382,386,394,398,
+A342193 ,1,15,33,35,45,51,55,69,75,77,85,91,93,95,99,105,119,123,135,141,143,145,153,155,161,165,175,177,187,195,201,203,205,207,209,215,217,219,221,225,231,245,247,249,253,255,265,275,279,285,287,291,295,297,299,
+A342194 ,1,1,1,3,3,5,7,7,7,13,11,11,17,13,15,25,17,17,29,19,23,35,25,23,39,29,29,45,33,29,55,31,35,55,39,43,65,37,43,65,51,41,77,43,51,85,53,47,85,53,65,87,61,53,99,67,67,97,67,59,119,61,71,113,75,79,123,67,79,117,
A342195 ,0,1,1,-5,-8,61,130,-1385,-3680,50521,160816,-2702765,-10026368,199360981,844583440,-19391512145,-92369507840,2404879675441,12722897618176,-370371188237525,-2154662195222528,69348874393137901,440001333689382400,-15514534163557086905,-106615331831035289600,4087072509293123892361,
A342196 ,1,1,5,23,155,1355,14371,183911,2781283,48726355,976903875,22183097191,565060532965,16016170519017,501714014484813,17265124180702953,649178961366102597,26544344366333824055,1175291769917975444817,56133021061270139242637,2881893164859601701738005,
A342197 ,1,1,9,63,919,18919,505639,18602319,877402487,51212704151,3688010412503,321523601578079,33283248550719793,4050897039400696253,574469890816237292037,93943844587040615104177,17565329004174205621822169,3730161837629377369026433019,
@@ -342065,6 +342099,7 @@ A342256 ,2,3,4,5,6,7,8,9,10,11,13,14,16,17,18,19,20,21,22,23,25,26,27,29,31,32,3
A342257 ,1,2,3,2,5,3,7,2,3,5,11,1,13,7,1,2,17,3,19,5,7,11,23,1,5,13,3,1,29,1,31,2,1,17,1,1,37,19,13,1,41,7,43,1,1,23,47,1,7,5,1,13,53,3,11,1,19,29,59,1,61,31,1,2,1,1,67,17,1,1,71,1,73,37,1,
A342258 ,62,74,188,194,195,275,278,363,398,422,423,483,494,495,614,662,663,747,758,764,782,867,1028,1071,1094,1095,1235,1238,1268,1394,1419,1454,1658,1659,1682,1844,1910,1916,1955,1970,2043,2067,2138,2139,2223,2235,2247,2259,
A342259 ,65,104,129,164,194,272,284,314,344,384,398,464,524,608,614,626,662,692,734,758,824,968,1025,1094,1172,1238,1280,1304,1364,1424,1448,1454,1532,1544,1595,1658,1664,1682,1724,1754,1832,1868,1869,1934,1952,2000,2001,2012,
+A342260 ,3,31,217,268,8399,29110,711243,4676815,31622764,376863606,12638826343,38121744938,1511790122972,
A342262 ,1,2,3,4,5,6,7,8,9,10,12,20,24,30,36,40,50,60,70,80,90,100,102,110,111,112,120,132,135,140,144,150,200,210,216,220,224,240,300,306,312,315,360,400,432,480,500,510,540,550,600,612,624,630,700,735,800,900,1000,1002,1008,
A342263 ,0,0,0,1,1,1,1,2,2,1,2,1,1,1,2,3,3,2,2,1,2,3,2,2,2,1,2,2,2,2,3,4,4,3,2,2,3,2,2,2,2,2,4,3,2,3,2,3,3,2,2,2,2,3,3,2,2,2,2,2,3,3,4,5,5,4,3,3,3,2,2,2,3,4,3,2,3,2,2,3,3,2,3,2,4,5,3,
A342264 ,0,1,2,3,4,5,6,7,8,9,13,11,12,14,15,18,16,17,19,25,22,23,24,33,26,29,27,28,38,39,49,66,45,34,35,44,55,56,57,58,59,67,46,68,47,69,48,77,36,78,37,79,88,89,99,123,111,112,113,114,115,118,116,117,119,125,122,124,133,126,129,127,128,138,
@@ -342103,6 +342138,7 @@ A342296 ,1,1,9,113,1649,2655,440985,7711009,138792929,
A342297 ,1,1,2,2,3,2,4,5,2,6,5,7,8,2,9,5,7,11,10,12,13,2,14,5,7,16,10,17,13,15,19,18,20,21,2,22,5,7,24,10,25,13,15,27,18,20,29,23,30,26,28,32,31,33,34,2,35,5,7,37,10,38,13,15,40,18,20,42,23,43,26,28,45,31,46,34,36,48,
A342298 ,2,25,931,504455,67539587599,585462196329239562271,21690980800898420269408456526391711768639,14792097944732868603877386771665610972834204784426907551800717772696470224928895,
A342299 ,41,281,827,857,2081,2801,8087,20981,21191,21491,81197,88607,206411,225941,227531,233141,249131,255971,261971,279551,283571,825107,827537,828407,834857,857567,861977,864047,869777,879167,883577,895787,2051111,2125601,2128601,2130701,2141801,2147021,2163221,
+A342302 ,6,12,48,90,252,294,300,420,432,720,798,864,930,1020,1140,1218,1368,1428,1602,1716,1890,1938,2088,2184,2190,2196,2250,2760,2880,3588,3660,3708,3774,3810,4452,4710,4902,5280,5340,5412,5754,5850,6174,6240,6462,6768,7014,7182,7632,8322,8820,9144,
A342303 ,1,1,0,3,0,5,6,4,0,0,7,14,0,16,10,15,13,0,21,0,39,10,58,8,0,49,16,81,68,36,49,72,0,39,33,25,25,0,40,16,11,106,6,7,0,9,10,26,60,85,11,70,40,9,214,30,32,52,16,0,65,30,6,226,0,24,130,161,20,0,99,0,68,216,136,0,62,26,129,
A342304 ,1,2,3,4,5,6,7,8,9,21,23,25,27,29,41,43,45,47,49,61,63,65,67,69,81,83,85,87,89,101,104,107,110,111,112,113,114,115,116,117,118,119,122,125,128,131,134,137,140,141,142,143,144,145,146,147,148,149,152,155,158,161,
A342306 ,1,0,0,0,240,0,20160,0,0,0,319334400,0,77127879628800,
@@ -342120,10 +342156,14 @@ A342317 ,0,2,6,9,44,60,35,234,564,504,135,1144,3816,6112,4080,527,5430,23000,511
A342318 ,1,1,1,1,1,5,1,61,1,1385,1,50521,691,2702765,1,199360981,3617,19391512145,43867,2404879675441,174611,370371188237525,77683,69348874393137901,236364091,15514534163557086905,657931,4087072509293123892361,3392780147,1252259641403629865468285,
A342319 ,1,2,12,56,120,992,252,16256,240,261632,132,4192256,32760,67100672,12,1073709056,8160,17179738112,14364,274877382656,6600,4398044413952,276,70368735789056,65520,1125899873288192,12,18014398375264256,3480,288230375614840832,
A342320 ,0,1,5,17,41,53,125,161,293,341,377,485,881,1025,1133,1313,1457,1805,2057,2393,2645,3077,3401,3941,4373,5333,5417,6173,6497,7181,7937,9197,9233,10205,11825,12641,13121,14153,14405,16001,16253,16757,18521,19493,21545,
+A342321 ,1,0,1,0,-1,2,0,1,-4,3,0,-3,22,-33,12,0,1,-13,33,-26,5,0,-5,114,-453,604,-285,30,0,5,-200,1191,-2416,1985,-600,35,0,-35,2470,-21465,62476,-78095,42930,-8645,280,0,14,-1757,21912,-88234,156190,-132351,51128,-7028,126,
+A342322 ,1,0,1,0,-1,2,0,0,-1,1,0,1,1,-9,6,0,0,1,1,-4,2,0,-1,-1,6,6,-15,6,0,0,-2,-2,5,5,-9,3,0,3,3,-17,-17,25,25,-35,10,0,0,3,3,-7,-7,7,7,-8,2,0,-5,-5,28,28,-38,-38,28,28,-27,6,0,0,-10,-10,23,23,-21,-21,12,12,-10,2,
A342323 ,1,1,1,1,2,1,1,1,3,1,1,2,1,2,1,1,1,1,1,5,1,1,2,3,2,1,1,1,1,1,1,1,1,3,7,1,1,2,1,2,1,1,1,2,1,1,1,3,1,1,1,1,1,3,1,1,2,1,2,5,3,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,11,1,1,2,3,2,1,1,1,2,3,1,1,1,1,
A342324 ,1,1,1,4,5,12,16,36,81,
A342325 ,174999,4,187,1,274,11,990213634,320741,108,59,16972551346,98100646316,
A342326 ,0,4,16,81,471,2031,1381,11781,6906,17956,34531,123256,40056,305256,863281,448906,200281,1957231,520731,10563906,1001406,11222656,7631406,3454506,1482081,75865156,7172606106,8852431,25035156,334020781,13018281,38531031,7410406,7014160156,
+A342327 ,64705,2542687,87880249,2867519047,91094247025,2857310964847,89080092692329,2769052985833687,85954322576134945,2666290098653287807,82680590830861862809,2563482326383161959527,79473712585542654112465,2463771499324688282695567,
+A342328 ,1068475,89633839,6458329435,433976684431,28211055010555,1804746233554159,114556965257054875,7243790885015626831,457188176014823960635,28828588756092946562479,1816999192589895468925915,114495695622871975031439631,
A342329 ,2,90,356232,152505051772,6961765466482521226,
A342330 ,1,1,2,2,3,4,4,7,9,11,17,23,32,44,63,91,127,180,255,363,516,732,1044,1485,2109,3002,4277,6089,8660,12323,17550,24986,35562,50628,72084,102616,146077,207980,296114,421555,600153,854469,1216543,1731983,2465842,3510713,
A342331 ,1,1,1,3,2,2,5,4,3,9,6,4,14,9,8,22,15,11,37,24,21,58,40,30,95,67,53,157,114,85,264,187,147,428,315,244,732,527,410,1207,892,681,2034,1490,1155,3416,2508,1927,5731,4215,3259,9597,7091,5454,16175,11914,9194,27134,20033,15425,45649,33672,25967,76714,
@@ -342138,6 +342178,7 @@ A342339 ,1,2,3,4,5,6,7,8,9,11,12,13,16,17,18,19,21,23,24,25,27,29,31,32,36,37,41
A342340 ,1,1,2,4,6,9,17,24,41,67,109,173,296,469,781,1284,2109,3450,5713,9349,15422,25351,41720,68590,112982,185753,305752,503041,827819,1361940,2241435,3687742,6068537,9985389,16431144,27036576,44489533,73205429,120460062,198214516,326161107,
A342341 ,1,1,1,1,1,3,1,3,3,5,5,5,9,7,13,15,17,19,29,31,39,43,63,59,75,121,119,169,167,199,279,305,343,479,537,733,789,883,1057,1421,1545,1831,2409,2577,3343,4001,4657,5131,6065,7755,8841,10473,12995,14659,17671,20619,25157,28255,33131,38265,47699,53171,62611,80005,88519,105937,119989,
A342342 ,1,1,1,3,1,3,5,5,3,11,9,11,17,15,29,39,31,39,65,57,107,127,149,155,187,265,293,419,523,571,781,763,941,1371,1387,2125,2383,2775,3243,4189,4555,
+A342343 ,1,1,1,3,3,5,8,10,13,18,27,32,44,55,73,97,121,151,194,240,299,384,465,576,706,869,1051,1293,1572,1896,2290,2761,3302,3973,4732,5645,6759,7995,9477,11218,13258,15597,18393,21565,25319,29703,34701,40478,47278,54985,
A342344 ,0,0,2,3,1,3,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,2,
A342345 ,3,33,363,36663,6306036,63066666036,6304963866683694036,6304963866689998999866683694036,
A342346 ,4,44,484,48884,8408048,84088888048,8408888888888888048,
@@ -342149,9 +342190,13 @@ A342351 ,11881,11882,11883,11884,11885,11886,11887,11888,11889,11890,11891,11892
A342352 ,0,0,0,0,4,15,41,98,218,465,967,1980,4016,8099,16277,32646,65398,130917,261971,524096,1048364,2096919,4194049,8388330,16776914,33554105,67108511,134217348,268435048,536870475,1073741357,2147483150,4294966766,8589934029,
A342353 ,0,8,80,416,1512,4216,10000,21256,
A342354 ,1,3,7,9,5,17,19,11,15,31,33,21,13,29,49,51,35,23,27,47,71,73,53,37,25,45,69,97,99,75,55,39,43,67,95,127,129,101,77,57,41,65,93,125,161,163,131,103,79,59,63,91,123,159,199,201,165,133,105,81,61,89,121,157,197,241,243,203,167,135,107,83,87,119,155,195,239,287,
+A342355 ,3,1,9,9,9,9,9,8,7,3,8,4,9,0,0,8,2,6,7,5,7,5,8,3,9,3,0,2,6,5,5,6,5,4,7,9,4,1,0,9,0,6,5,1,4,9,2,0,8,2,9,3,9,6,9,6,4,0,9,9,0,9,6,6,9,6,3,1,9,5,7,6,8,4,6,6,0,8,3,2,2,1,1,7,1,2,9,5,9,5,8,9,1,8,4,9,0,
A342356 ,1,10,12,2,20,22,24,4,14,16,6,26,28,8,18,15,5,25,35,30,3,33,36,32,34,38,48,40,42,21,27,57,45,50,52,54,44,46,56,58,68,60,62,64,66,63,39,9,69,90,70,7,77,147,49,84,74,37,333,93,31,124,72,75,51,17,102,80,78,76,86,82,88,98,91,
A342357 ,1,2,11,125,1469,30970,1424807,25646168,943532049,66190291008,1883023236995,119209289551407,8338590851427689,366451025462807402,25231464507361789935,2996947275258886238380,211289282287835811874277,12680220578500976681544666,1815313698001596651227722787,
A342358 ,1,6,140,270,2970,332640,14303520,5297292000,
+A342359 ,6,4,5,4,7,5,2,4,4,5,6,5,0,0,3,9,2,4,4,3,5,7,3,1,5,5,4,5,6,6,0,6,6,3,6,5,2,2,4,6,7,7,2,0,5,5,9,4,0,2,1,5,1,6,1,8,1,6,8,0,0,6,7,5,3,1,7,5,0,9,5,5,3,7,3,1,2,5,6,8,8,3,6,5,1,3,9,2,5,3,9,2,7,1,9,0,
+A342360 ,4,0,7,1,7,6,3,8,7,2,9,6,5,6,7,1,5,7,9,0,2,8,9,0,2,0,4,7,3,5,3,9,7,6,7,7,3,1,0,5,1,0,6,4,4,1,3,4,5,2,8,4,6,5,1,4,4,9,3,3,3,9,6,9,2,9,8,1,3,2,0,9,6,6,7,5,4,1,8,5,8,6,9,5,0,8,4,0,5,5,0,8,9,6,6,6,
+A342361 ,1,3,0,9,6,8,9,0,0,5,6,6,3,4,5,6,0,0,8,5,8,0,7,5,4,3,3,6,9,5,6,3,7,0,4,8,4,2,2,6,4,2,9,6,1,5,5,6,4,7,3,1,8,4,3,0,5,9,6,7,0,0,9,6,2,9,1,2,9,0,0,7,5,5,4,0,2,1,6,9,2,6,1,3,0,8,0,3,5,0,0,6,8,6,1,1,
A342362 ,1,10,31,76,145,254,399,600,849,1170,1551,2020,2561,3206,3935,4784,5729,6810,7999,9340,10801,12430,14191,16136,18225,20514,22959,25620,28449,31510,34751,38240,41921,45866,50015,54444,59089,64030,69199,74680,80401,86450,92751,99396,106305,113574,
A342363 ,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,3,19,3,3,3,19,3,3,3,19,
A342364 ,3,17,73,191,709,1289,3181,5449,7681,17477,33889,87961,437389,2290573,7160227,10429681,19196227,24504049,47577857,70513979,82605937,156671243,271785793,328939937,568119509,1125978241,1534657963,1710749497,4936728373,7647104183,
@@ -342175,6 +342220,7 @@ A342382 ,0,1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,19,18,17,20,21,23,26,24,27,25,29,
A342383 ,0,1,2,3,4,5,7,6,8,9,10,13,12,14,15,16,18,17,19,20,21,24,23,25,26,27,29,28,30,31,32,35,34,36,37,38,40,39,41,42,43,46,45,47,48,49,53,50,52,51,54,69,56,64,59,61,62,58,65,60,63,57,67,68,70,72,71,74,73,75,78,76,80,79,81,82,83,
A342384 ,0,1,1,1,1,2,0,4,6,4,0,2,
A342385 ,0,1,1,2,1,2,3,2,2,3,4,3,4,3,4,5,4,6,6,4,5,6,5,8,9,8,5,6,7,6,10,12,12,10,6,7,8,7,12,15,16,15,12,7,8,9,8,14,18,20,20,18,14,8,9,10,9,16,21,24,25,24,21,16,9,10,11,10,18,24,28,30,30,28,24,18,10,11,
+A342386 ,2,5,23,5,13,7,7,79,37,23,67,89,131,31,71,47,43,73,277,353,41,67,127,223,79,13,193,5,23,43,5,67,3,19,5,59,59,653,19,19,97,409,5,383,29,137,379,349,653,1187,47,41,37,17,619,89,283,283,43,479,191,1009,571,
A342387 ,20,175,1500,29600,253075,1124039,2163720,1620864179,3120083460,13857908224,118481007099,2337285022799,19983094049524,170849530073079,28815607761506104,127985053235771120,246364903884373539,1094234263598927875,184554358010701244300,1577885049278315692375,
A342388 ,6,55,474,9360,80029,355452,684228,512562258,986657022,4382255359,37466984190,739114421304,6319209189385,54027365220036,9112295268838531,40472427468293976,77907423180308442,346027256676968725,58361212342395772530,498971064164650006699,4266054677084570952198,
A342389 ,1,5,30,264,3135,46709,823564,16777528,387420759,10000003265,285311670666,8916100500148,302875106592331,11112006826381965,437893890380965260,18446744073726350224,827240261886336764313,39346408075296928032645,
@@ -342192,10 +342238,11 @@ A342400 ,1,3,3,3,3,12,3,3,3,15,3,13,3,15,15,3,3,15,3,13,15,15,3,15,3,15,3,15,3,7
A342401 ,1,2,6,10,30,42,60,66,78,90,110,130,170,190,210,330,390,462,510,546,570,690,798,858,870,930,1050,1110,1218,1230,1290,1410,1470,1554,1590,1722,1770,1830,1974,2010,2130,2190,2310,2730,3570,3990,4290,4830,5610,6006,
A342402 ,36,100,144,324,400,576,784,900,1296,1600,1764,1936,2304,2500,2704,2916,3136,3600,4356,4624,4900,5184,5776,6084,6400,7056,7744,8100,9216,9604,10000,10404,10816,11025,11664,12100,12544,12996,14400,15876,16900,17424,18496,
A342403 ,1,-1,-1,1,-1,4,-1,-3,2,6,-1,-24,-1,8,7,21,-1,-38,-1,-58,9,12,-1,288,4,14,-16,-108,-1,-180,-1,-315,13,18,11,930,-1,20,15,1126,-1,-314,-1,-256,-116,24,-1,-6960,6,-154,19,-354,-1,1078,15,2940,21,30,-1,6664,-1,32,-198,9765,17,
+A342404 ,0,0,0,0,6,40,165,546,1596,4320,11115,27610,66858,158808,371553,858690,1964280,4454272,10024407,22410234,49803750,110096280,242216205,530573890,1157621556,2516575200,5452587075,11777596506,25367140386,54492386200,116769410745,
A342405 ,0,45,2268,76221,2245320,62858025,1723364748,46836754821,1268169391440,34282547074305,926123262507828,25011175461289821,675371104361586360,18235844869321055385,492377645105637260508,13294313813660319607221,358947876218708733778080,
A342406 ,11,13,19,23,31,41,53,59,61,67,79,89,103,139,167,179,193,199,241,251,257,277,347,367,373,409,461,463,467,479,523,541,563,601,613,641,653,691,719,743,811,823,853,881,887,937,947,977,1039,1063,1087,1117,1129,1151,1223,1249,1259,1277,1283,
A342407 ,0,5,51,498,5004,50028,500014,4999954,49999325,499998777,5000002329,49999998413,499999949299,4999999991841,49999999683763,499999999022579,5000000005362272,50000000022520652,500000000055534895,5000000000274296550,50000000000909149240,
-A342408 ,2,3,53,53,523,6337,36947,36947,277363,8177791,8622017,8622017,565337239,3361495721,
+A342408 ,2,3,53,53,523,6337,36947,36947,277363,8177791,8622017,8622017,565337239,3361495721,16747915297,76675792867,76675792867,633679985683,633679985683,4443195645419,21685290410821,205793034752197,
A342409 ,9,21,166,317,596,4167,26448,48970,90652,302042,
A342411 ,1,2,7,34,501,2600,100843,1048610,28697821,400000502,23579476911,247669459528,21505924728445,340163474352620,15569560546875507,576460752304472098,45798768824157052689,728637186579594211070,98646963440126439346903,
A342412 ,1,2,7,37,501,2771,100843,1056833,28702189,401562757,23579476911,247792605523,21505924728445,340246521979079,15569565432876147,576478345026355201,45798768824157052689,728648310343004595593,98646963440126439346903,
@@ -342216,6 +342263,7 @@ A342426 ,1,2,6,9,14,21,40,42,56,72,84,108,110,120,126,130,143,154,156,162,165,16
A342427 ,1,168,459,1817,2196,2197,2655,3128,3280,3699,4199,4575,4927,5184,5795,6600,7215,7259,7656,7657,8448,9636,11304,11339,12492,14160,14175,14424,14805,15624,15625,16335,16336,16925,17802,19170,20349,20811,21624,21735,22197,
A342428 ,2196,7656,15624,16335,64375,109224,171624,202824,328887,329427,392733,393640,447578,482238,494450,520695,631824,723519,773790,785695,820960,876987,981783,986607,1021824,1026750,1030455,1084048,1108094,1160670,1235070,1242824,1412908,
A342429 ,1649373,4029519,15281054,31906263,43387386,58198173,94468958,100084949,131393766,131986502,140282279,156786124,211004079,246960048,253000850,278206663,310135917,330168203,351204398,363280904,412296883,504736647,515831624,537255647,566300238,
+A342430 ,0,1,1,2,1,12,5,108,145,974,2210,17073,31950,238591,
A342431 ,1,2,5,8,13,18,21,24,29,31,34,38,42,46,
A342432 ,1,2,5,22,129,1411,16813,266372,4787349,100391653,2357947701,61980047702,1792160394049,56707753687079,1946197516142925,72061992621375496,2862423051509815809,121441389759089405193,5480386857784802185957,
A342433 ,1,3,11,74,629,8085,117655,2113796,43059849,1001955177,25937424611,743379914746,23298085122493,793811662313709,29192938251553759,1152956691126550536,48661191875666868497,2185928270773974154773,
@@ -342231,6 +342279,7 @@ A342442 ,2,3,4,5,6,7,8,9,42,14,17,18,15,16,13,19,32,22,23,26,29,12,25,24,34,27,3
A342443 ,5,97,991,9949,99971,999983,9999991,99999989,999999937,9999999943,99999999977,999999999989,9999999999763,99999999999959,
A342444 ,2,3,5,9,5,29,281,1575,599,7,17,3,6449,2725361,
A342445 ,22,33,44,48,55,66,77,88,99,122,124,126,155,162,168,184,202,204,222,244,248,264,280,288,303,324,330,333,336,366,396,404,408,412,420,424,440,444,448,488,505,515,555,606,636,648,660,666,707,728,770,777,784,808,824,840,
+A342446 ,1,4,1,9,2,1,21,3,1,1,44,4,1,1,1,90,6,2,1,1,1,182,9,2,1,1,1,1,367,13,3,1,1,1,1,1,736,19,3,1,1,1,1,1,1,1475,27,4,1,1,1,1,1,1,1,2952,38,5,2,1,1,1,1,1,1,1,5907,54,6,2,1,1,1,1,1,1,1,1,
A342447 ,1,1,1,1,1,1,3,1,1,4,8,2,1,1,4,11,29,12,5,1,1,4,12,43,105,92,45,12,3,1,1,4,12,46,156,460,582,487,204,71,14,7,1,1,4,12,47,170,670,2097,3822,4514,3271,1579,561,186,44,16,4,1,1,4,12,47,173,731,2954,10513,24584,40182,
A342448 ,1,3,7,10,18,25,30,36,52,67,80,94,103,113,125,136,168,199,228,258,283,309,337,364,381,399,419,438,462,485,506,528,592,655,716,778,835,893,953,1012,1061,1111,1163,1214,1270,1325,1378,1432,1465,1499,1535,1570,
A342449 ,1,5,29,262,3129,46705,823549,16777544,387421251,10000003469,285311670621,8916100581446,302875106592265,11112006826387025,437893890391180013,18446744073743123788,827240261886336764193,39346408075299116257065,
@@ -342263,6 +342312,8 @@ A342477 ,1,1,2,1,1,1,3,2,1,1,1,2,1,1,3,1,5,2,1,1,1,2,6,1,3,1,2,1,1,7,1,2,1,3,1,1
A342478 ,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,20,21,24,28,30,32,33,35,36,39,40,42,44,45,48,51,52,55,56,57,60,63,91,117,126,133,171,182,189,217,234,247,252,259,266,273,275,279,341,451,550,671,682,775,781,825,902,
A342479 ,0,1,1,1,46,44,288,33216,613248,151296,391584768,2383570944,86830424064,206470840320,21270238986240,987259950858240,1262040231444480,3022250536693923840,3884253754215628800,1102040800033347993600,1892288242221318144000,5616902226049109065728000,
A342480 ,1,6,10,15,1155,1365,12155,1616615,37182145,11849255,33426748355,247357937827,10141675450907,25652473199353,2928046583754721,155186468939000213,223317113839049087,558516101711461766587,796182527971658263007,241532826894674874877669,430046252763689411367557,
+A342482 ,0,12,50,150,392,952,2214,5010,11132,24420,53066,114478,245520,524016,1113806,2358954,4980356,10485340,22019634,46136838,96468440,201325992,419429750,872414530,1811938572,3758095572,7784627354,16106126430,33285995552,68719475680,
+A342483 ,0,30,150,525,1568,4284,11070,27555,66792,158730,371462,858585,1964160,4454136,10024254,22410063,49803560,110096070,242215974,530573637,1157621280,2516574900,5452586750,11777596155,25367140008,54492385794,116769410310,249644959665,
A342484 ,4,4,4,8,2,2,1,6,1,5,2,6,0,5,
A342485 ,1,2,17,34,4097,146,1679617,262178,60466193,4198402,1000000000001,67109042,1283918464548865,470186664194,281474976714769,2251799813947426,4722366482869645213697,609359800476818,12748236216396078174437377,9223372036858974242,
A342486 ,1,7,8,2,6,6,1,9,2,1,6,2,7,8,9,7,7,0,3,1,8,1,0,4,0,7,6,1,8,9,3,3,6,2,9,6,9,9,5,2,1,2,4,5,6,4,9,1,2,9,2,0,7,7,4,3,9,8,9,5,9,7,4,5,9,6,5,8,6,5,3,1,3,2,3,7,0,9,5,5,8,5,7,1,5,5,3,7,8,4,1,4,7,2,8,4,1,7,8,0,
@@ -342311,7 +342362,7 @@ A342528 ,1,1,2,4,7,12,20,32,51,79,121,182,272,399,582,839,1200,1700,2394,3342,46
A342529 ,1,1,2,3,7,13,19,36,67,114,197,322,564,976,1614,2729,4444,7364,12357,20231,33147,
A342530 ,1,2,2,3,2,6,2,6,3,6,2,12,2,6,6,9,2,12,2,12,6,6,2,28,3,6,6,12,2,26,2,14,6,6,6,31,2,6,6,28,2,26,2,12,12,6,2,52,3,12,6,12,2,28,6,28,6,6,2,66,2,6,12,25,6,26,2,12,6,26,2,76,2,6,12,12,6,26,
A342531 ,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,2,1,1,0,1,0,0,1,2,1,1,1,1,0,1,0,0,1,1,2,2,1,1,1,0,1,0,0,1,1,2,3,1,1,1,1,0,1,0,0,
-A342532 ,1,0,1,2,3,4,9,14,28,44,83,136,250,424,757,1310,2313,4018,7081,12314,21650,37786,66264,115802,
+A342532 ,1,0,1,2,3,4,9,14,28,44,83,136,250,424,757,1310,2313,4018,7081,12314,21650,37786,66264,115802,202950,354858,621525,1087252,1903668,3330882,5831192,10204250,17862232,31260222,54716913,95762576,167614445,293356422,513456686,
A342534 ,1,2,6,7,20,12,42,26,50,40,110,42,156,84,120,100,272,100,342,140,252,220,506,156,484,312,438,294,812,240,930,392,660,544,840,350,1332,684,936,520,1640,504,1806,770,1000,1012,2162,600,2022,968,1632,1092,2756,876,2200,1092,2052,
A342535 ,1,2,10,11,68,20,222,78,238,136,1010,110,1740,444,680,604,4112,476,5850,748,2220,2020,10670,780,8276,3480,6330,2442,21980,1360,27030,4792,10100,8224,15096,2618,46692,11700,17400,5304,64040,4440,74130,11110,16184,21340,97382,6040,
A342536 ,1,1,3,4,10,17,36,65,126,227,419,743,1323,2295,3965,
@@ -342323,6 +342374,8 @@ A342541 ,1,2,4,5,8,10,12,14,28,28,20,62,24,54,272,68,32,198,36,676,1224,130,44,1
A342542 ,1,2,3,4,5,7,7,9,15,13,11,33,13,19,105,33,17,91,19,209,469,31,23,641,1045,37,1627,841,29,4217,31,673,10461,49,29785,10281,37,55,49465,68769,41,65197,43,12281,529625,67,47,273185,279979,1049661,1049121,52657,53,803647,
A342543 ,1,2,10,19,1028,76,279942,65558,10077718,1049608,100000000010,16777334,106993205379084,78364444044,35184372090920,281474976776236,295147905179352825872,101559966746268,708235345355337676357650,1152921504607897676,46005119909369702026044,10000000000100000000020,
A342544 ,1,2,6,11,260,40,46662,16398,1679630,262408,10000000010,4194366,8916100448268,13060740684,4398046511640,35184372105244,18446744073709551632,16926661124436,39346408075296537575442,144115188076118572,3833759992447475215524,1000000000010000000020,
+A342545 ,2,24,16,280,216,3430,4096,19683,100000,4348377,2985984,154457888,105413504,4442343750,4294967296,313909084845,198359290368,8712567840033,10240000000000,500396429346030,584318301411328,38112390316557080,36520347436056576,298023223876953125,
+A342546 ,3,7,73,141,1417,17130,11677,187955,10252371,20440221,1550384575,10645648530,80224807014,829050923579,17071371319785,
A342550 ,2,2,2,5,2,2,5,6,2,5,2,2,10,6,2,5,2,2,11,6,2,5,7,2,5,13,2,5,2,2,5,6,7,19,2,2,5,6,2,5,2,2,19,6,2,5,9,2,11,6,2,5,17,2,5,6,2,5,2,2,5,14,7,22,2,2,5,13,2,5,2,2,10,6,2,17,2,2,20,6,2,5,7,2,5,
A342551 ,1,4,9,8,16,32,27,25,64,128,81,72,512,1024,108,2048,243,49,4096,8192,16384,288,729,32768,125,225,200,131072,262144,2187,524288,1152,1048576,432,2097152,4194304,972,196,8388608,648,33554432,4608,864,67108864,19683,268435456,
A342552 ,2,3,4,3,5,6,9,11,8,12,24,19,34,14,27,14,28,17,46,26,24,55,28,14,86,50,38,66,28,67,76,41,64,40,43,93,53,87,67,48,89,66,42,72,69,76,49,76,42,49,59,73,260,109,145,169,70,137,193,292,
@@ -342332,6 +342385,7 @@ A342570 ,0,1,1,3,4,9,13,29,46,101,167,375,644,1461,2563,5899,10534,24469,44237,1
A342571 ,1,7,9,8,0,7,9,7,4,3,4,1,0,4,7,7,3,4,2,1,5,2,4,5,4,9,5,9,0,4,3,9,6,3,8,8,2,0,4,2,6,5,9,3,5,0,6,0,0,7,3,9,8,3,9,3,1,0,3,2,3,4,8,7,8,1,2,8,3,0,6,7,3,4,6,6,7,3,3,5,5,7,3,3,3,9,2,
A342572 ,1,3,5,7,9,15,17,21,25,27,31,35,45,49,51,63,73,75,81,85,93,105,107,119,125,127,135,147,153,155,175,189,217,219,225,243,245,255,257,279,289,313,315,321,343,357,365,375,381,405,425,441,443,459,465,511,525,527,
A342573 ,1,2,24,5184,39813120,17915904000000,702142910300160000000,3330690501757390081228800000000,2534703826002712645182542460223488000000000,395940866122425193243875570782668457763038822400000000000,
+A342574 ,9,2,5,3,5,8,3,5,6,2,3,6,0,4,0,6,3,3,3,7,0,8,8,4,1,6,6,3,7,0,7,6,3,8,2,8,0,4,9,5,6,5,0,1,5,9,9,1,6,1,0,7,2,8,7,1,0,4,0,7,1,4,8,5,1,7,8,6,7,9,5,3,3,0,7,3,1,8,5,8,4,4,4,4,9,3,2,9,8,8,5,2,1,0,3,6,8,6,8,7,4,4,6,0,3,7,
A342575 ,4,5,6,7,14,15,26,102,103,104,224,103,104,105,506,507,452,1169,1170,1171,8228,10419,15186,5227,16619,16620,16621,25102,130090,62640,330791,330792,351403,273100,681504,649069,352375,3045104,3045105,3635007,9532211,7819691,3091425,3091426,
A342576 ,1,4,4,4,5,8,13,14,14,16,22,24,29,33,
A342577 ,1,3,4,6,9,11,11,13,15,19,16,20,25,25,27,27,29,35,30,34,41,41,39,41,47,45,44,48,57,53,57,55,57,67,56,62,73,71,67,69,73,79,68,76,89,83,87,83,93,89,86,90,105,99,95,97,109,99,100,104,121,109,117,111,
@@ -342340,6 +342394,7 @@ A342579 ,5,10,17,23,23,24,34,39,39,45,46,71,71,71,71,95,95,95,95,95,95,95,95,96,
A342581 ,7,13,14,16,31,31,31,32,55,55,55,55,55,61,62,64,69,74,81,127,127,127,127,127,127,127,127,127,127,127,127,128,133,138,145,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,223,
A342582 ,2,2,12,12,10,12,42,56,558,10,682,12,52,42,150,240,170,558,38,240,42,682,598,240,150,52,3132,56,232,150,558,992,8382,170,2730,936,666,38,936,240,738,42,3010,3784,535230,598,11938,240,2254,150,204,52,212,3132,
A342583 ,3,6,18,42,82,271,284,369,445,682,1069,1193,1900,2241,3894,6137,7108,8164,9658,10126,12645,14842,14936,17913,18420,19480,23893,24605,28959,32913,36279,40847,43936,44559,45500,
+A342584 ,7,25,627,2454,136120,376847,2886750,21546984,278567575,2437795018,97974268952,4836489478578,4836489478578,147895359776636,308788493220129,4193528956200935,25999253094360135,650467164953053602,2161492060929047665,26769019461318409710,
A342586 ,1,63,6087,608383,60794971,6079301507,607927104783,60792712854483,6079271032731815,607927102346016827,60792710185772432731,6079271018566772422279,607927101854119608051819,60792710185405797839054887,6079271018540289787820715707,607927101854027018957417670303,
A342587 ,1,1,2,1,12,6,1,86,108,24,1,840,2310,960,120,1,11642,65700,42960,9000,720,1,227892,2583126,2510760,712320,90720,5040,1,6285806,142259628,199357704,71310960,11481120,987840,40320,1,243593040,11012710470,21774014640,9501062760,1781015040,
A342588 ,1,0,2,0,0,12,0,0,0,128,18,0,0,0,0,2000,960,100,0,0,0,0,0,41472,43320,15000,1710,140,0,0,0,0,0,0,1075648,1985760,1453200,490560,90594,10080,770,0,0,0,0,0,0,0,33554432,96937680,122360000,82220880,32527488,8205288,1396640,179760,20048,1050,
@@ -342355,12 +342410,14 @@ A342597 ,1,2,4,6,6,8,9,11,12,14,15,18,17,22,21,22,25,26,27,31,31,32,33,34,38,39,
A342598 ,1,8,216,19584,5542200,4551802560,10225942680240,
A342599 ,1,3,12,74,788,16016,658220,
A342601 ,10,224,278,286,452,473,502,510,645,656,698,744,871,889,909,921,955,960,966,972,1010,1062,1086,1113,1121,1163,1182,1200,1201,1208,1271,1273,1282,1315,1327,1328,1377,1431,1444,1510,1541,1550,1564,1570,1583,1610,1626,1630,1674,1677,1693,1706,1719,1720,1726,1738,
+A342602 ,0,0,1,1,1,4,6,14,29,63,129,300,756,1677,4134,9525,22841,57175,141819,354992,882420,2218078,5588989,14173217,35918542,
A342603 ,0,1,1,7,1,13,7,43,1,19,13,85,7,85,43,259,1,25,19,127,13,163,85,517,7,127,85,553,43,517,259,1555,1,31,25,169,19,241,127,775,13,241,163,1063,85,1027,517,3109,7,169,127,847,85,1063,553,3361,43,775,517,3361,259,3109,1555,9331,1,
A342604 ,2,5,10,17,39,52,69,126,195,224,255,403,649,821,868,921,1216,1826,2496,2851,2924,3003,3501,4836,6776,8291,8909,9016,9125,9916,12583,17168,21963,24882,25925,26076,26233,27537,32213,41901,54431,64567,69915,71459,71656,71855,73754,81782,100850,129704,
A342605 ,1,2,4,14,20,26,31,39,42,57,64,69,87,92,114,127,150,152,172,213,274,301,326,379,436,460,499,523,597,708,747,817,819,912,1382,1452,1595,1600,1603,1632,1647,1670,1768,1833,1834,1873,1890,1986,2137,2696,2702,2859,3080,3154,3167,3173,3386,3933,
A342606 ,2,5,17,821,2851,8291,12583,32213,64567,193283,481409,507979,2889443,3026911,15233891,24430993,95635361,95754697,221519339,1147397567,12921783863,28711457651,63521027291,305580335353,1449155675287,3157505489581,6839699592347,14717764856273,67875393766057,
A342607 ,1,2,2,3,2,11,2,19,66,1027,2,835,2,279939,1052674,69635,2,10114563,2,1074855939,78364426242,100000000003,2,4315152387,1099511627778,106993205379075,101559973445634,21937029021319171,2,1162183941554179,2,562950221856771,10000000000001073741826,
A342608 ,1,2,65,258,1048577,4610,78364164097,4294971394,101559956672513,1100585369602,10000000000000000000001,281474977071106,11447545997288281555215581185,6140964151415455875074,1237940039285381374411014145,79228162514264619068521709570,
+A342609 ,17,49,161,197,199,209,251,391,419,449,649,685,769,799,883,967,1057,1189,1249,1301,1457,1481,1681,1793,1937,1979,2001,2029,2089,2177,2209,2311,2377,2379,2419,2431,2449,2549,2551,2575,2591,2705,2729,2899,3041,3073,
A342610 ,0,1,5,6,25,11,30,31,125,36,55,41,150,61,155,156,625,161,180,91,275,96,205,191,750,211,305,216,775,311,780,781,3125,786,805,341,900,271,455,366,1375,371,480,301,1025,396,955,941,3750,961,1055,516,1525,521,1080,991,
A342611 ,0,1,7,8,49,15,56,57,343,64,105,71,392,113,399,400,2401,407,448,169,735,176,497,463,2744,505,791,512,2793,799,2800,2801,16807,2808,2849,855,3136,617,1183,904,5145,911,1232,673,3479,960,3241,3207,19208,3249,3535,1296,5537,
A342612 ,1,2,5,10,257,50,46657,16450,1679681,327682,10000000001,4196098,8916100448257,15237476354,4398063289345,35184640528386,18446744073709551617,19747769389058,39346408075296537575425,
@@ -342372,9 +342429,15 @@ A342617 ,1,2,3,4,5,20,7,38,135,4102,11,1670,13,1679624,4202505,270346,17,6053991
A342618 ,1,2,129,514,4194305,9218,470184984577,17179877378,609359740018689,4402341478402,100000000000000000000001,1125899907563522,137370551967459378662586974209,36845784908492735250434,9903520314283046597240029185,
A342619 ,1,2,9,18,1025,98,279937,65666,10077825,1310722,100000000001,16780802,106993205379073,91424858114,35184439199745,281476050460674,295147905179352825857,118486616186882,708235345355337676357633,1152921796664688642,46005120518729441509377,11000000000000000000002,
A342620 ,1,2,33,66,16385,770,10077697,1050626,362805249,83886082,10000000000001,268664834,15407021574586369,19747769352194,2252074693689345,18014673389486082,75557863725914323419137,25593109118189570,229468251895129407139872769,73788172563556335618,6624765697237267477692417,11000000000000000000000002,
+A342621 ,0,2,3,4,7,5,15,6,6,9,56,7,101,17,10,8,297,8,490,11,18,58,1255,9,14,103,9,19,4565,12,6842,10,59,299,22,10,21637,492,104,13,44583,20,63261,60,13,1257,124754,11,30,16,300,105,329931,11,63,21,493,4567,831820,
+A342622 ,0,0,1,3,5,7,7,6,4,4,5,8,12,16,21,26,30,33,33,32,30,26,23,23,24,28,30,33,38,38,37,35,29,21,14,6,-3,-12,-21,-29,-38,-47,-54,-60,-61,-63,-68,-71,-78,-82,-88,-88,-87,-85,-77,-68,-58,-47,-35,
+A342623 ,0,1,2,2,1,-1,-4,-7,-10,-14,-18,-21,-23,-26,-27,-29,-33,-38,-44,-50,-56,-61,-67,-74,-81,-87,-94,-101,-107,-115,-123,-131,-137,-140,-145,-149,-149,-148,-146,-141,-138,-134,-127,-119,-109,-99,-90,-80,
+A342624 ,0,1,2,2,2,0,-1,-2,0,1,3,4,4,5,5,4,3,0,-1,-2,-4,-3,0,1,4,2,3,5,0,-1,-2,-6,-8,-7,-8,-9,-9,-9,-8,-9,-9,-7,-6,-1,-2,-5,-3,-7,-4,-6,0,1,2,8,9,10,11,12,12,11,12,10,12,11,12,9,8,5,7,2,
+A342625 ,1,1,0,-1,-2,-3,-3,-3,-4,-4,-3,-2,-3,-1,-2,-4,-5,-6,-6,-6,-5,-6,-7,-7,-6,-7,-7,-6,-8,-8,-8,-6,-3,-5,-4,0,1,2,5,3,4,7,8,10,10,9,10,8,10,9,11,11,11,8,7,6,4,0,-1,-5,-2,-7,-3,-6,-4,-9,
A342628 ,1,2,2,6,2,45,2,322,731,3383,2,132901,2,827641,10297068,33570818,2,2578617270,2,44812807567,678610493340,285312719189,2,393061010002613,95367431640627,302875123369471,150094917726535604,569939345952661545,2,105474306078445349841,2,
A342629 ,1,3,10,69,626,7866,117650,2101265,43047451,1000390658,25937424602,743069105634,23298085122482,793728614541474,29192926269590300,1152925902670135553,48661191875666868482,2185913413229070900339,104127350297911241532842,
A342630 ,5,6,7,14,18,24,29,34,39,41,47,53,77,114,119,148,150,159,176,188,189,190,191,205,215,217,218,241,268,288,312,314,331,334,339,342,346,352,364,367,387,390,402,418,429,438,439,440,446,449,480,493,494,500,504,510,521,523,546,549,553,561,580,
+A342631 ,2,4,2,4,3,3,2,4,3,4,4,9,6,6,9,5,11,12,10,7,2,13,8,5,5,4,5,5,8,13,4,7,10,12,11,5,9,6,6,9,3,11,12,10,7,5,4,3,4,3,3,3,4,3,4,5,7,10,12,11,2,9,6,6,9,5,11,12,10,7,4,13,8,5,5,4,5,5,8,13,3,7,10,12,11,5,
A342632 ,1,3,11,43,159,647,2519,10043,39895,159703,637927,2551171,10200039,40803219,163198675,652774767,2611029851,10444211447,41776529287,167106121619,668423198491,2673693100831,10694768891659,42779072149475,171116268699455,684465093334979,2737860308070095,
A342633 ,0,1,1,4,1,7,4,13,1,10,7,25,4,25,13,40,1,13,10,37,7,46,25,79,4,37,25,88,13,79,40,121,1,16,13,49,10,67,37,118,7,67,46,163,25,154,79,241,4,49,37,136,25,163,88,277,13,118,79,277,40,241,121,364,1,19,16,61,13,88,49,157,
A342634 ,0,1,1,5,1,9,5,21,1,13,9,41,5,41,21,85,1,17,13,61,9,77,41,169,5,61,41,185,21,169,85,341,1,21,17,81,13,113,61,253,9,113,77,349,41,333,169,681,5,81,61,285,41,349,185,761,21,253,169,761,85,681,341,1365,1,25,21,101,17,
@@ -342419,7 +342482,9 @@ A342672 ,1,3,20,63,42,60,88,135,325,126,156,1260,238,264,840,2511,342,975,460,12
A342673 ,1,2,3,1,1,6,1,8,1,2,1,6,1,2,3,1,1,2,1,4,3,2,1,24,1,2,3,4,1,6,1,4,3,2,1,1,1,2,3,40,1,6,1,2,1,2,1,6,7,2,3,26,1,6,1,8,3,2,1,12,1,2,3,1,1,6,1,4,3,2,1,8,1,2,3,4,7,6,1,8,1,2,1,12,5,2,3,8,1,2,1,2,3,2,1,24,1,14,1,1,1,6,1,8,3,
A342674 ,1,1,2,36,1,2,5,120,1,4,2,4,336,19,2,36,8,4,264,1,2,24,30,56,8,1092,1,2,1,12,28,56,4,612,1,4,9,11,12,418,8,20,2280,1,6,2,10,1,48,26,8,20,5520,1,2,4,4,266,1,48,34,24,40,6960,1,2,180,4,42,308,1,12,76,24,60,1984,3,2,18,240,4,798,26,1,20,138,12,4,2812,1,2,
A342675 ,1,3,4,13,6,120,8,1161,2197,16148,12,603190,14,5773008,50422464,201359377,18,16590656229,20,269768284118,4748723771432,3138430473896,24,2972582195034162,476837158203151,3937376419253748,1350852564961601560,4066515044181860654,30,1036488835382356683530,32,
+A342676 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,
A342677 ,1,5,28,265,3126,46916,823544,16793633,387422677,10001953190,285311670612,8916464313700,302875106592254,11112103714568680,437893891601739648,18446779258148749825,827240261886336764178,39346424755299348744797,1978419655660313589123980,
+A342678 ,0,1,2,2,3,3,3,3,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,10,10,10,10,11,11,11,11,12,12,13,13,14,14,15,15,16,16,17,17,17,17,17,17,18,18,18,18,19,19,19,19,20,20,20,20,20,20,20,20,21,21,22,22,23,23,24,
A342680 ,9,6,1,3,9,4,3,1,5,9,4,5,7,3,6,5,4,7,2,4,7,6,4,5,9,5,3,1,6,1,5,4,7,3,0,6,8,6,8,5,8,2,6,9,3,0,1,0,5,8,4,6,0,4,5,5,1,1,5,1,4,9,1,8,1,8,6,3,3,7,8,0,2,9,1,4,6,9,9,7,0,6,6,7,5,4,2,4,3,2,5,5,4,9,5,5,5,5,2,6,9,8,7,9,2,
A342681 ,241,443,613,641,811,20011,20047,20051,20101,20161,20201,20347,20441,20477,21001,21157,21211,21377,21467,22027,22031,22147,22171,22247,22367,23017,23021,23131,23357,23417,23447,24007,24121,24151,24407,25031,25111,25117,25121,26021,26107,26111,26417,27011,27407,28001,
A342697 ,0,0,0,1,0,1,3,3,0,0,2,3,6,7,7,7,0,0,0,1,4,5,7,7,12,12,14,15,14,15,15,15,0,0,0,1,0,1,3,3,8,8,10,11,14,15,15,15,24,24,24,25,28,29,31,31,28,28,30,31,30,31,31,31,0,0,0,1,0,1,3,3,0,0,2,3,6,
@@ -342429,26 +342494,37 @@ A342700 ,0,0,2,0,7,0,0,0,15,6,10,0,3,0,0,0,31,14,30,12,23,4,16,0,7,6,2,0,3,0,0,0
A342701 ,3,7,5,14,9,34,7,16,15,26,11,68,39,28,15,32,33,72,25,40,35,56,17,101,45,37,45,56,29,152,31,61,39,56,35,144,37,61,39,74,41,128,35,88,45,161,47,192,49,82,51,74,95,216,43,97,75,203,59,304,91,88,63,122,
A342702 ,1,2,4,6,12,18,24,30,48,60,78,90,120,150,180,210,330,360,390,420,630,840,1050,1260,1470,1680,1890,2100,2310,3360,3570,3990,4200,4620,5460,6300,6930,9240,10710,10920,11550,13860,16380,17220,17850,18480,20790,27720,30030,39270,
A342703 ,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,3,2,4,2,5,2,6,2,7,2,8,2,9,3,4,3,5,3,6,3,7,3,8,3,9,4,5,4,6,4,7,4,8,4,9,5,6,5,7,5,8,5,9,6,7,6,8,6,9,7,8,7,9,8,9,1362,1816,1635,1962,2334,2723,3366,3927,
+A342704 ,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,1,
A342705 ,5,7,13,17,19,59,97,101,107,109,191,223,229,277,283,569,613,631,643,709,719,743,829,857,881,1031,1049,1051,1091,1109,1171,1193,1249,1277,1301,1327,1489,1579,1637,1697,1949,1979,2003,2081,2089,2113,2141,2203,2357,2423,2539,2593,2659,2749,2789,2819,
A342706 ,13,31,79,109,151,1201,3271,3469,3889,4111,12289,16879,17791,25951,27673,108301,126079,134857,138679,169957,174259,186019,231877,245389,259309,355009,367501,371737,397489,412939,461017,477619,524197,544429,565069,602401,741031,833191,904303,961069,1267501,
+A342708 ,1,0,1,0,-1,1,0,1,-1,1,0,-1,0,-1,1,0,1,0,0,-1,1,0,-1,1,-1,0,-1,1,0,1,-1,2,-1,0,-1,1,0,-1,0,-1,1,-1,0,-1,1,0,1,0,1,0,1,-1,0,-1,1,0,-1,1,-2,1,-1,1,-1,0,-1,1,0,1,-1,2,-1,2,-1,1,-1,0,-1,1,
A342709 ,1,64,3025,142129,6677056,313679521,14736260449,692290561600,32522920134769,1527884955772561,71778070001175616,3372041405099481409,158414167969674450625,7442093853169599697984,349619996931001511354641,16424697761903901433970161,
A342710 ,3,18,123,843,5778,39603,271443,1860498,12752043,87403803,599074578,4106118243,28143753123,192900153618,1322157322203,9062201101803,62113250390418,425730551631123,2918000611027443,20000273725560978,137083915467899403,939587134549734843,
+A342711 ,1,3,6,9,13,17,22,27,32,38,44,50,57,64,71,78,86,94,102,110,119,128,137,146,155,165,175,185,195,205,216,227,238,249,260,271,283,295,307,319,331,343,356,369,382,395,408,421,434,448,462,476,490,504,518,532,547,
+A342712 ,0,0,0,1,2,4,6,9,13,17,22,28,34,41,49,58,67,77,88,100,112,125,139,154,170,186,203,221,240,260,280,301,323,346,370,395,420,446,473,501,530,560,590,621,653,686,720,755,791,827,864,902,941,981,1022,1064,1106,
A342713 ,2,9,21,54,90,144,234,350,504,714,950,1350,1764,2156,2772,3500,4374,5390,6380,7812,9504,10890,12740,14850,17442,20475,23100,26334,30444,34320,38709,43146,48510,55250,61047,66780,74925,83600,92169,100485,109350,121512,133331,144000,156195,171171,
+A342714 ,1,0,5,9,0,6,4,2,6,
A342715 ,5,7,13,15,15,20,23,26,31,31,39,41,41,47,49,52,57,57,62,65,68,73,75,81,83,83,89,91,94,99,99,107,109,109,115,117,123,125,125,130,133,136,141,143,149,151,151,157,159,162,167,167,172,175,178,183,185,191,193,
A342716 ,3,16,19,42,42,42,55,58,76,79,79,110,110,110,118,121,144,144,144,155,160,173,181,181,207,207,207,220,223,254,254,254,275,275,275,283,283,309,309,309,320,325,343,346,346,377,377,377,385,388,406,409,409,422,
+A342717 ,7,13,139,1049,4481,8147,11047,11411,13049,17191,17921,25913,26321,28057,30169,33349,37561,38177,40487,42139,60493,65563,72871,74507,74521,77041,77069,93491,112363,127849,130621,138389,142787,144577,145109,158227,161561,165311,
A342718 ,0,1,2,1,3,3,2,1,1,1,4,1,3,4,4,1,2,1,2,3,2,2,2,1,1,1,2,1,5,2,2,1,2,1,2,1,3,5,5,1,3,2,3,5,4,2,2,1,3,1,2,1,3,5,2,5,4,1,3,5,3,2,1,1,3,2,2,3,2,2,2,1,4,1,1,1,2,3,2,1,1,1,
A342719 ,21,36,45,55,78,78,78,120,136,120,105,171,210,210,171,136,231,300,325,300,231,171,300,406,465,465,406,300,210,378,528,630,666,630,528,378,253,465,666,820,903,903,820,666,465,300,561,820,1035,1176,1225,1176,1035,820,561,
+A342720 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,3,1,2,2,3,1,4,2,4,2,5,3,7,1,2,4,3,13,7,20,12,5,3,7,10,3,8,2,14,12,10,15,17,8,11,10,20,13,15,10,45,9,18,25,46,38,18,2,25,20,30,18,32,17,32,43,
+A342721 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,2,0,0,0,3,1,1,0,0,1,3,0,0,0,2,1,0,6,0,4,4,2,1,0,0,1,0,0,6,0,2,8,6,2,0,1,2,0,2,0,9,0,0,2,0,13,1,0,4,0,3,0,3,5,10,11,
+A342724 ,1,1,2,1,3,1,1,3,4,2,5,3,4,3,2,0,4,5,5,6,5,2,5,3,4,5,5,6,9,6,5,7,10,7,9,6,6,7,9,6,7,4,6,7,6,6,9,10,10,11,10,7,12,10,9,9,8,8,11,11,11,12,12,10,13,9,11,12,11,7,9,10,13,14,13,10,12,11,10,
A342725 ,0,1,13,17,189,205,257,273,3005,3069,3277,3341,4033,4097,4305,4369,48061,48317,49149,49405,52173,52429,53261,53517,64449,64705,65537,65793,68561,68817,69649,69905,768957,769981,773309,774333,785405,786429,789757,790781,834509,
A342726 ,1,2,3,4,5,6,7,10,12,15,16,18,20,24,25,30,32,33,35,36,40,42,44,45,48,50,54,60,64,65,66,70,77,80,88,90,96,99,100,110,112,120,124,125,126,130,140,144,145,147,150,156,160,168,170,180,182,184,185,186,190,192,
A342727 ,2,21,26,31,36,41,46,51,310,315,325,330,335,340,345,350,355,360,365,370,375,390,395,405,410,415,420,425,430,435,455,470,475,485,490,495,535,550,555,565,570,575,580,585,590,595,600,605,610,620,625,630,635,645,
A342728 ,0,1,2,3,4,5,6,7,23,39,55,71,87,103,359,615,871,1127,1383,1639,5735,9831,13927,18023,22119,26215,91751,157287,222823,288359,353895,419431,1468007,2516583,3565159,4613735,5662311,6710887,23488103,40265319,57042535,73819751,
A342729 ,1,3,5,7,9,22,24,26,39,41,43,56,58,60,73,75,77,90,92,94,107,109,111,136,138,140,153,155,157,170,172,174,199,201,203,216,218,220,233,235,237,262,264,266,279,281,283,296,298,300,313,315,317,330,332,334,347,349,
+A342730 ,2,5,7,8,20,20,22,19,40,42,36,70,66,57,49,94,88,73,129,116,99,85,149,135,120,197,172,149,121,206,196,165,271,236,211,172,291,256,216,175,309,262,223,364,316,263,219,392,335,273,445,390,325,268,459,395,
A342731 ,0,2,3,4,4,4,5,8,10,14,5,5,6,5,5,16,11,6,15,6,9,11,6,6,10,16,15,6,6,11,17,16,6,11,6,10,16,6,6,11,10,6,12,6,7,7,7,10,16,11,7,12,16,6,7,6,7,11,12,6,18,7,8,18,10,7,12,10,7,7,11,12,17,9,7,12,
A342732 ,3,7,14,17,19,35,37,49,51,60,66,70,74,80,82,89,91,103,155,161,163,170,172,184,218,224,225,226,230,233,235,238,256,266,270,273,277,297,300,304,318,322,326,328,330,336,340,357,363,367,372,376,382,398,404,
A342733 ,2,5,6,9,11,12,15,16,18,20,23,24,26,27,28,29,30,32,34,36,38,41,42,44,45,46,47,50,53,54,55,56,57,59,61,63,65,68,69,71,72,73,75,77,78,79,81,83,84,85,86,87,88,90,92,93,96,97,99,101,104,105,107,108,
A342734 ,1,4,8,10,13,21,22,25,31,33,39,40,43,48,52,58,62,64,67,76,94,95,98,100,102,106,112,116,118,121,130,143,144,145,148,149,152,154,157,175,176,179,181,183,187,193,197,199,202,211,216,219,223,229,237,241,247,
A342735 ,1,2,4,5,9,10,12,13,15,19,20,24,25,26,31,32,34,35,38,40,41,43,44,50,52,54,55,57,58,59,63,64,68,70,71,75,76,77,82,83,87,88,90,91,93,94,96,100,102,103,104,108,109,111,114,117,118,120,121,126,127,129,
A342736 ,3,6,7,8,11,14,16,17,18,21,22,23,27,28,29,30,33,36,37,39,42,45,46,47,48,49,51,53,56,60,61,62,65,66,67,69,72,73,74,78,79,80,81,84,85,86,89,92,95,97,98,99,101,105,106,107,110,112,113,115,116,119,122,
+A342737 ,19,71,181,379,701,1189,1891,2861,4159,5851,8009,10711,14041,18089,22951,28729,35531,43471,52669,63251,75349,89101,104651,122149,141751,163619,187921,214831,244529,277201,313039,352241,395011,441559,492101,546859,606061,669941,738739,812701,892079,
A342738 ,5,7,19,41,197,2549,4159,8467,9433,26701,27551,46817,57037,91097,130859,153281,157049,197683,351727,423103,466181,517991,526291,567181,575231,652903,663167,772339,1055231,1062013,1088239,1171199,1232461,1551871,1603297,1662833,2782469,2920531,2957917,3226159,
A342739 ,1,1,2,2,2,3,2,3,3,3,4,3,3,4,4,3,4,5,4,4,4,3,5,5,4,4,5,4,6,5,4,5,5,5,4,6,4,6,5,4,5,6,5,5,7,5,6,5,4,6,6,5,6,5,6,7,5,5,7,6,5,5,6,5,7,6,4,6,8,6,6,7,6,6,5,5,7,7,5,6,7,6,6,7,5,8,
A342741 ,2,5,7,9,12,15,16,20,22,24,26,28,32,35,37,38,41,44,46,49,50,54,57,59,61,63,67,68,71,73,75,77,79,83,85,87,90,93,96,97,102,104,106,108,111,112,114,115,117,120,124,126,129,132,133,137,139,141,146,147,
@@ -342456,7 +342532,16 @@ A342742 ,1,4,10,13,19,25,31,34,40,43,52,55,58,64,70,76,82,88,91,94,100,103,109,1
A342743 ,7,16,22,28,37,46,49,61,67,73,79,85,97,106,112,115,124,133,139,148,151,163,172,178,184,190,202,205,214,220,226,232,238,250,256,262,271,280,289,292,307,313,319,325,334,337,343,346,352,361,373,379,388,397,
A342744 ,3,6,8,11,14,17,18,21,23,27,29,30,33,36,39,42,45,47,48,51,53,56,60,62,65,66,69,72,74,78,80,81,84,86,89,92,95,98,99,101,105,107,110,113,116,119,122,123,125,128,131,134,135,138,140,143,144,149,150,152,
A342745 ,2,5,9,12,15,20,24,26,32,35,38,41,44,50,54,57,59,63,68,71,75,77,83,87,90,93,96,102,104,108,111,114,117,120,126,129,132,137,141,146,147,155,158,161,164,168,170,173,174,177,182,188,191,195,200,201,207,210,
+A342746 ,2,3,5,8,12,6,9,21,35,17,26,14,20,11,15,62,102,48,75,39,57,30,44,33,54,27,41,23,32,18,24,183,143,116,170,89,129,98,161,80,120,66,93,53,71,95,155,74,114,60,87,47,68,51,83,42,63,36,50,29,38,197,156,
+A342747 ,2,1,7,5,4,22,3,16,13,12,67,10,9,49,40,8,37,202,35,31,28,6,148,121,26,25,112,21,607,106,20,94,85,102,19,445,17,364,79,15,76,337,75,64,62,319,61,14,283,
+A342748 ,0,1,1,0,2,2,1,1,3,1,3,2,0,2,4,2,2,4,2,3,1,1,3,5,1,3,3,3,5,3,1,4,2,3,2,4,2,6,2,0,4,4,2,4,6,4,4,2,2,5,3,2,4,3,4,5,3,2,7,3,3,1,5,1,5,3,1,5,7,3,5,5,5,3,3,1,6,4,3,3,5,3,4,5,3,6,
+A342749 ,1,1,2,1,1,2,2,2,1,3,2,3,1,2,1,2,4,2,3,3,2,2,2,1,3,3,4,2,2,4,3,3,2,3,3,2,4,1,4,1,3,4,3,3,2,2,4,4,2,3,2,5,4,3,3,2,5,3,1,4,4,2,3,3,4,4,2,3,2,3,3,4,2,4,3,3,3,2,4,6,4,5,3,4,2,2,
+A342750 ,1,2,4,5,9,10,12,13,15,19,24,34,35,38,40,41,52,55,57,59,76,88,90,93,102,104,114,121,130,136,137,142,145,147,182,207,208,211,228,241,248,260,284,294,305,312,316,328,338,350,355,364,370,376,406,430,432,
+A342751 ,1,2,6,8,9,10,12,13,16,18,22,23,24,27,30,32,33,34,39,41,43,44,45,46,48,52,54,56,57,58,59,61,62,64,65,72,75,76,77,78,79,81,84,85,88,92,94,96,101,102,104,105,106,107,108,109,111,112,113,115,121,122,
+A342752 ,3,4,5,7,11,14,15,17,19,20,21,25,26,28,29,31,35,36,37,38,40,42,47,49,50,51,53,55,60,63,66,67,68,69,70,71,73,74,80,82,83,86,87,89,90,91,93,95,97,98,99,100,103,110,114,116,117,118,119,120,123,124,128,
+A342753 ,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,
A342755 ,2,3,4,5,6,7,8,9,42,15,22,14,55,12,37,16,25,36,29,47,23,46,13,44,18,32,17,38,19,33,26,35,174,53,76,59,34,27,43,67,49,62,87,106,493,57,24,75,48,65,122,39,54,72,88,45,66,73,56,77,52,79,84,63,78,123,69,58,64,92,74,68,114,85,314,
+A342756 ,28,63,125,213,362,489,696,823,1104,1443,1642,2089,2433,2616,3060,3555,4103,4396,5072,5477,5792,6550,7033,7781,8614,9342,9749,10258,10773,11449,13173,13814,14682,15433,16669,17262,18248,19363,20269,
A342757 ,9,12,17,14,22,28,17,27,37,42,19,32,45,55,59,22,37,54,68,78,79,24,42,62,81,96,104,102,27,47,71,94,115,129,135,128,29,52,79,107,133,154,167,169,157,32,57,88,120,152,179,200,210,208,189,34,62,96,133,170,204,232,251,258,250,224,
A342758 ,12,15,23,19,30,37,22,37,48,54,26,44,60,71,74,29,51,71,88,97,97,33,58,83,105,121,128,123,36,65,94,122,144,159,162,152,40,72,106,139,168,190,202,201,184,43,79,117,156,191,221,241,250,243,219,47,86,129,173,215,252,281,299,303,290,257,
A342759 ,1,2,3,4,6,10,16,25,43,73,133,241,457,865,1681,3265,6433,12673,25153,49921,99457,198145,395521,789505,1577473,3151873,6300673,12595201,25184257,50356225,100700161,
@@ -342465,21 +342550,32 @@ A342761 ,4,7,10,15,25,43,79,147,283,547,1075,2115,4195,8323,16579,33027,65923,13
A342762 ,1,1,1,1,1,3,5,10,20,42,86,178,362,738,1490,3010,6050,12162,24386,48898,97922,196098,392450,785410,1571330,3143682,6288386,
A342763 ,1,2,2,2,3,6,10,15,27,45,85,153,297,561,1105,2145,4257,8385,16705,33153,66177,131841,263425,525825,1051137,2100225,4199425,8394753,16787457,33566721,67129345,134242305,268476417,536920065,1073823745,2147581953,4295131137,
A342764 ,1,1,2,4,7,13,26,51,97,191,366,713,1375,2673,5164,10031,19405,37663,72922,141461,274019,531405,1029640,1996395,3868793,7500411,14536342,28179521,54617039,
+A342765 ,1,2,2,3,2,3,4,3,3,4,5,4,3,4,5,6,5,6,6,5,6,7,6,5,4,5,6,7,8,7,6,10,10,6,7,8,9,8,7,6,5,6,7,8,9,10,9,12,14,10,10,14,12,9,10,11,10,9,8,7,6,7,8,9,10,11,12,11,10,9,20,14,14,20,9,10,11,12,
+A342766 ,1,2,3,6,10,10,14,28,42,42,66,66,78,78,78,156,204,204,228,228,228,228,276,276,460,460,690,690,870,870,930,1860,1860,1860,1860,1860,2220,2220,2220,2220,2460,2460,2580,2580,2580,2580,2820,2820,3948,3948,3948,3948,
+A342767 ,1,1,1,1,2,1,1,2,2,1,1,4,3,4,1,1,2,4,4,2,1,1,4,3,8,3,4,1,1,2,6,4,4,6,2,1,1,8,3,8,5,8,3,8,1,1,4,8,4,6,6,4,8,4,1,1,4,9,16,5,12,5,16,9,4,1,1,2,6,8,8,6,6,8,8,6,2,1,1,8,3,8,9,16,7,16,9,8,3,8,1,
+A342768 ,1,2,3,8,5,12,7,32,27,20,11,48,13,28,45,128,17,108,19,80,63,44,23,192,125,52,243,112,29,180,31,512,99,68,175,432,37,76,117,320,41,252,43,176,405,92,47,768,343,500,153,208,53,972,275,448,171,116,59,720,
A342769 ,1,1,1,3,2,2,1,5,2,4,3,3,1,7,2,6,3,5,4,4,1,9,2,8,3,7,4,6,5,5,1,11,2,10,3,9,4,8,5,7,6,6,1,13,2,12,3,11,4,10,5,9,6,8,7,7,1,15,2,14,3,13,4,12,5,11,6,10,7,9,8,8,1,17,2,16,3,15,4,14,5,13,6,12,7,
A342770 ,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,2,0,1,0,1,0,0,3,0,1,0,1,0,5,0,3,0,1,0,1,0,0,7,0,3,0,1,0,1,0,14,0,8,0,3,0,1,0,1,0,0,22,0,8,0,3,0,1,0,1,0,42,0,24,0,8,0,3,0,1,0,1,0,0,66,0,25,0,8,0,
A342771 ,43,53,79,103,227,769,977,1303,2179,2803,3019,5179,5503,8089,8101,10651,10789,13339,13729,14419,16069,17053,17341,18077,23203,25111,26153,26161,32839,34127,34351,34519,38791,39103,44027,54319,56629,57503,59053,60811,62869,63079,64579,64591,65203,69019,
A342772 ,3,4,5,9,7,8,13,13,11,17,13,14,24,20,17,25,19,20,32,27,23,33,29,26,37,37,29,41,31,32,45,41,40,56,37,38,53,48,41,57,43,44,70,55,47,65,53,50,74,62,53,73,65,56,77,69,59,81,61,62,85,80,73,98,67,68,93,88,71,97,
A342773 ,2,4,8,17,18,25,38,72,118,121,161,234,245,275,329,347,521,614,720,830,944,998,1016,1318,1355,1664,1829,2041,2169,2183,2189,2384,2786,3115,3464,3710,4082,4472,4891,4900,5027,5315,6230,6543,6836,7889,8173,10190,10592,10601,11435,11858,12154,12752,
+A342774 ,1,2,1,3,2,2,4,3,3,3,5,2,4,4,4,4,6,3,3,5,3,5,5,5,5,7,4,4,4,4,6,4,4,6,6,6,6,6,3,8,5,5,5,5,5,5,7,5,5,5,5,7,7,7,7,7,4,7,4,4,9,6,6,6,6,4,6,6,6,6,8,6,6,6,6,6,6,6,8,8,8,8,8,5,8,5,
A342800 ,0,0,0,0,0,0,24,72,0,0,1704,5184,0,0,193344,600504,0,0,34321512,141520752,0,0,9205815672,37962945288,0,0,
A342801 ,127,173,181,257,281,359,563,569,577,719,751,1061,1381,1879,1901,4327,4759,5441,6397,6977,7207,7933,8387,8419,8521,9349,10009,10891,11311,11443,11467,12323,13567,13873,14369,14929,15299,15683,16073,17351,18041,18749,24407,24481,24767,25819,27067,27739,
+A342802 ,0,1,-3,-2,9,10,6,7,-27,-26,-30,-29,-18,-17,-21,-20,81,82,78,79,90,91,87,88,54,55,51,52,63,64,60,61,-243,-242,-246,-245,-234,-233,-237,-236,-270,-269,-273,-272,-261,-260,-264,-263,-162,-161,-165,-164,-153,-152,-156,-155,-189,-188,-192,-191,-180,-179,-183,-182,
+A342803 ,13,67,449,1367,1230127,4004009,121200307,10022234347,10203242527,52281509069,90608667517,100200322224127,121022023024027,9659504223792743,
+A342804 ,0,0,1,1,1,5,8,18,39,91,185,460,1051,2526,6280,15645,35516,93765,225989,611503,
A342805 ,1,1,1,3,1,1,4,3,1,5,1,1,18,4,1,7,1,1,24,5,1,9,4,1,10,18,1,11,1,1,12,7,5,156,1,1,14,8,1,15,1,1,288,9,1,17,4,1,90,10,1,19,21,1,20,11,1,21,1,1,22,48,8,414,1,1,24,65,1,25,1,1,234,14,1,81,1,1,784,15,
A342806 ,15,21,78,300,528,903,990,1830,2628,3240,3570,4278,5253,5460,7503,8778,9870,13203,13530,16653,18528,20100,22578,24753,25200,29403,31878,37128,39903,45753,48828,55278,64980,65703,72390,73920,81003,88410,98790,106953,107880,
A342807 ,1,6,30,150,750,3750,18630,92406,458262,2270478,11245590,55697766,275769654,1365260862,6758345838,33450929886,165549052326,819248589606,4054005363918,
A342808 ,1,2,6,4,8,28,14,48,55,98,154,54,495,1034,504,559,208,440,2078,2000,350,3519,6578,2574,5983,2924,21734,25023,11934,30303,120175,81718,11438,73150,71630,43470,50048,511784,371448,37960,1478048,391950,812174,393470,217854,576288,
+A342809 ,8,12,14,24,54,84,114,234,264,294,354,444,504,564,654,684,744,864,954,984,1164,1194,1284,1554,1584,1734,1914,2004,2154,2214,2244,2334,2394,2544,2844,2964,3084,3204,3414,3594,
A342810 ,1,2,3,4,5,6,7,8,9,21,27,81,191,243,729,999,2187,2997,6561,8991,19683,26973,33321,36963,39049,59049,80919,100389,110889,118827,177147,177897,183951,242757,332667,356481,531441,551853,728271,998001,1069443,1367631,1594323,1655559,2184813,
A342811 ,1,13,1009,354161,496376001,2632501072321,52080136110870785,3872046158193220660993,1099175272489026844687825921,1210008580962784935280673680079873,5225407816779297641534116390319222362113,
A342812 ,1,1,7,142,5895,417201,45046558,6891812712,1417730229765,377158121463025,
A342813 ,4,3,1,4,0,7,1,2,5,4,6,6,7,7,2,9,5,0,3,3,0,2,2,9,1,9,8,6,4,1,6,3,0,9,3,7,3,0,0,9,2,6,6,3,4,2,2,4,7,6,6,2,7,8,6,3,6,5,4,4,0,3,7,7,7,2,9,8,2,9,0,3,4,1,7,4,0,3,6,3,9,6,1,3,1,3,4,
+A342814 ,12,14,18,38,68,98,158,308,338,368,398,488,548,758,788,908,968,998,1118,1568,1658,1748,1868,1988,2288,2438,2618,2708,2858,2888,3038,3068,3218,3308,3458,3548,3638,3698,3848,4058,
+A342815 ,3,13,53,213,227,853,909,3413,3637,13653,14549,14563,54613,58197,58253,218453,232789,233013,873813,931157,932053,932067,3495253,3724629,3728213,3728269,13981013,14898517,14912853,14913077,55924053,59594069,59651413,59652309,
A342817 ,1,-4,4,16,52,112,-48,-1984,-11212,-33360,6224,713536,4441872,13004480,-17374656,-432012032,-2525831628,-6454496208,21147389392,326358047552,1794285832464,4124461926592,-19727734694848,-263598020446976,-1416694290412784,-3151402998261312,
A342818 ,2,2,8,2,6,8,8,2,10,6,4,8,4,8,20,2,18,10,5,6,8,4,6,8,6,4,11,8,5,20,32,2,34,18,4,10,7,5,9,6,7,8,8,4,10,6,6,8,6,6,9,4,7,11,9,8,13,5,9,20,6,32,68,2,66,34,6,18,12,4,9,10,6,7,9,5,8,9,11,6,
A342819 ,4,4,7,6,9,10,6,11,12,13,8,13,16,17,16,8,15,18,21,20,19,10,17,22,25,26,25,22,10,19,24,29,30,31,28,25,12,21,28,33,36,37,36,33,28,12,23,30,37,40,43,42,41,36,31,14,25,34,41,46,49,50,49,46,41,34,14,27,36,45,50,55,56,57,54,51,44,37,
@@ -342497,9 +342593,12 @@ A342834 ,7,797,797997,7979979973,797997997399991,797997997399991999983,797997997
A342835 ,2,2,8,16,8,4,64,4,32,8,128,64,16,64,512,8,8,64,
A342836 ,7,797,3,7,37,3023681,43,1249,7,3,23,11,3,19,3,13390093693131976661567,193,2069,11,41,3,71,3,996370591,3,101,1123,54367,159469,151,29,3,7,
A342837 ,0,0,3,3,16,40,8,44,112,85,48,24,168,15,182,18,13,151,348,204,437,612,771,75,51,310,796,111,811,350,644,350,469,159,571,544,2239,4,1474,97,2177,175,1400,1791,75,1983,337,2503,854,2397,830,246,5350,1682,153,1581,622,
+A342838 ,21,31,32,39,42,62,67,75,82,91,93,97,104,109,121,127,135,137,139,140,145,146,
A342839 ,1,4,7,9,10,15,16,22,23,24,25,34,36,37,39,40,47,55,56,57,58,64,67,82,84,86,87,88,91,93,94,95,96,97,98,99,100,102,104,105,106,107,130,133,134,135,136,137,138,139,140,141,142,144,146,147,148,149,150,153,
A342840 ,1,1,2,6,23,1,103,10,6,1,512,77,69,30,21,5,6,2740,548,598,330,335,123,174,58,58,37,26,3,9,1,15485,3799,4686,2970,3411,1676,2338,1040,1317,878,777,363,608,230,252,165,133,30,93,26,31,4,1,3,4,91245,26165,35148,24550,30182,17185,24685,12976,16867,12248,12360,7203,11086,5692,6391,5194,5006,2751,3917,2019,2482,1622,1371,812,1233,490,495,416,360,157,282,54,78,41,29,22,49,7,4,0,6,
A342841 ,1,841,832693,832046137,831916552903,831908477106883,831907430687799769,831907383078281024371,831907373418800027750413,831907372722449100147414487,831907372589073124899487831735,831907372581823023465031521920149,831907372580768386561159867257319711,
+A342843 ,0,4,3,4,5,3,6,7,4,3,9,6,10,6,3,4,
+A342844 ,27,34,38,46,49,54,56,57,58,68,69,74,76,78,86,87,94,98,203,207,209,247,249,253,259,267,289,299,308,323,329,334,338,343,346,356,358,370,374,376,377,380,386,388,394,398,403,406,407,429,430,434,437,446,447,454,
A342845 ,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,2,1,1,1,2,1,2,1,2,1,1,0,1,0,1,0,1,0,1,0,2,1,2,1,1,1,2,1,2,1,1,0,1,0,1,0,1,0,1,0,2,1,2,1,2,1,1,1,2,1,1,0,1,0,1,0,1,0,1,0,2,1,2,1,2,1,2,1,1,1,1,0,1,0,1,0,1,0,1,0,2,2,3,2,3,
A342846 ,0,0,0,0,0,0,0,0,0,1,1,1,2,1,2,1,2,1,2,0,1,0,1,0,1,0,1,0,1,1,2,1,1,1,2,1,2,1,2,0,1,0,1,0,1,0,1,0,1,1,2,1,2,1,1,1,2,1,2,0,1,0,1,0,1,0,1,0,1,1,2,1,2,1,2,1,1,1,2,0,1,0,1,0,1,0,1,0,1,1,2,1,2,1,2,1,2,1,1,1,1,1,2,1,2,
A342847 ,1,1,1,2,3,2,3,4,6,5,4,7,2,5,3,4,5,11,3,6,2,11,4,3,2,14,5,12,4,14,9,18,11,17,5,24,8,12,25,28,11,26,19,14,27,12,18,5,2,34,22,32,26,9,17,29,23,12,43,6,47,4,16,32,16,4,16,30,9,12,57,37,29,28,
@@ -342508,10 +342607,11 @@ A342849 ,0,0,1,2,-3,-8,-9,944,-6336,-27745,-565504,-436807,-57869312,123505175,-
A342851 ,0,1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,61,62,63,64,65,66,67,68,69,71,72,73,74,75,76,77,78,79,81,82,
A342852 ,2,5,5,7,8,12,33,52,93,236,479,1265,2782,6650,15539,
A342853 ,0,0,0,0,1,3,6,13,24,42,68,106,153,217,300,
-A342854 ,0,0,0,0,1,2,5,9,17,26,41,60,88,120,163,
+A342854 ,0,0,0,0,1,2,5,9,17,26,41,60,88,120,163,213,
A342855 ,10,20,30,40,50,60,70,80,90,100,101,102,104,105,110,120,140,150,200,202,204,208,210,220,240,250,280,300,303,306,330,360,400,404,408,420,440,480,500,505,510,520,540,550,600,606,630,660,700,707,770,800,808,840,880,900,909,990,1000,
A342856 ,1,24,120,720,5040,40320,362880,3628800,39916800,479001600,6227020800,87178291200,1307674368000,20922789888000,355687428096000,6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000,
A342858 ,13530,136,35,5,4510,10,100,45,51,1404,
+A342859 ,0,1,1,2,1,2,3,2,2,3,4,3,3,3,4,5,4,5,5,4,5,6,5,8,7,8,5,6,7,6,11,11,11,11,6,7,8,7,14,17,17,17,14,7,8,9,8,17,24,26,26,24,17,8,9,10,9,20,31,39,39,39,31,20,9,10,11,10,23,38,55,60,60,55,38,
A342860 ,1,1,2,6,23,1,103,9,8,512,62,82,34,28,2,2740,402,612,384,466,94,232,42,60,8,15485,2593,4187,3036,4356,1746,3132,1064,1918,909,654,333,612,144,104,22,24,1,91245,16921,28065,21638,33274,17598,31180,12942,24000,14290,15434,7770,15692,5965,6896,3947,5660,2226,3674,1314,1512,516,508,204,332,37,40,
A342861 ,1,1,2,6,23,1,103,10,6,1,513,75,74,26,17,9,6,2762,522,645,321,290,130,166,47,54,48,41,4,8,2,15793,3579,5023,3058,3232,1527,2228,874,1159,893,875,340,503,281,269,207,156,112,123,21,54,2,0,6,5,
A342862 ,1,1,2,6,23,1,103,11,4,2,513,88,53,33,18,8,6,0,0,1,2761,642,495,340,262,160,172,65,58,39,14,6,18,0,0,6,0,0,2,15767,4567,4099,3007,2692,1832,2171,1152,1291,968,728,457,566,174,176,221,129,14,122,29,38,52,8,0,32,9,0,10,0,0,8,0,0,0,0,0,1,
@@ -342522,6 +342622,8 @@ A342866 ,1,2,3,2,3,2,3,2,3,3,3,2,3,3,4,2,3,2,3,3,4,3,3,2,3,3,3,3,3,4,3,2,6,3,5,2
A342867 ,1,2,3,15,35,33,65,215,221,551,455,2001,3417,3621,11523,16705,16617,69845,107545,157285,324569,358883,1404949,1569295,3783970,3106285,7536065,12216295,10589487,24038979,57759065,51961945,177005465,131462695,741703701,1467144445,
A342868 ,1,4,8,16,32,64,112,128,224,256,448,896,1568,1792,3136,3584,5824,6272,7168,11648,12544,14336,23296,25088,28672,39424,40768,46592,50176,78848,81536,93184,128128,157696,163072,186368,256256,326144,372736,512512,652288,885248,1025024,
A342869 ,1,9,27,45,90,180,270,360,540,1080,2160,2700,4320,4860,5400,8100,9720,10800,16200,19440,29160,32400,48600,58320,64800,97200,129600,162000,165240,183600,194400,275400,291600,330480,367200,388800,550800,583200,660960,777600,972000,
+A342871 ,1,3,5,8,10,12,14,17,20,22,24,26,28,30,32,36,38,40,42,44,46,48,50,52,55,57,60,62,64,66,68,71,73,75,77,80,82,84,86,88,90,92,94,96,98,100,102,104,107,109,111,113,115,117,119,121,123,125,127,129,131,133,
+A342872 ,0,1,2,3,2,1,0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,
A342873 ,0,7,16,62,92,213,276,508,616,995,1160,1722,1956,2737,3052,4088,4496,5823,6336,7990,8620,10637,11396,13812,14712,17563,18616,21938,23156,26985,28380,32752,34336,39287,41072,46638,48636,54853,57076,63980,66440,74067,
A342874 ,293,2106,2161,2763,3698,3793,3812,3922,3959,4000,4205,4224,4260,4728,4953,5065,5283,5617,5700,5751,5932,6326,6333,6422,6539,6623,7375,7475,7501,7533,7542,8306,8568,8751,8777,8994,9102,9259,9354,9480,10389,10700,10791,
A342875 ,1,4,6,2,7,0,8,5,5,0,9,3,1,8,5,7,9,4,3,3,7,7,3,6,9,7,0,4,9,2,6,2,3,1,5,6,2,6,5,4,6,2,3,9,7,8,1,7,3,8,3,2,3,7,3,7,5,3,6,9,8,8,4,7,1,4,4,9,9,5,6,8,2,5,8,6,4,7,8,2,6,0,3,7,2,6,7,
@@ -342537,31 +342639,464 @@ A342885 ,1,18,306,5202,88146,1493874,25300530,428518386,7256300850,122876680626,
A342886 ,1,20,380,7220,136820,2593100,49121660,930556460,17625825740,333857601020,6323384122580,119767717450100,2268399952520660,42963566150826380,813721674662589980,15411746407417290020,291893918240586194660,5528387235193561980740,
A342887 ,1,22,462,9702,203302,4260542,89253582,1869809502,39167457582,820458452462,17185914925542,359989506212182,7540511273930822,157947298263243742,3308420553034902382,69299392385043268822,1451565583054963249302,30404929596858248780502,
A342888 ,1,24,552,12696,291480,6692424,153614760,3526063752,80931227016,1857565708968,42634594787160,978544945823832,22459264078075992,515478463349872200,11831064537706447464,271542137952854806776,6232321082672399260152,143041632747658763159736,
-A342907 ,1,4,20,304,6784,407684,39072966,9449433606,
+A342889 ,1,1,1,1,11,1,1,66,66,1,1,286,1716,286,1,1,1001,26026,26026,1001,1,1,3003,273273,1184183,273273,3003,1,1,8008,2186184,33157124,33157124,2186184,8008,1,1,19448,14158144,644195552,2254684432,644195552,14158144,19448,1,
+A342890 ,1,1,1,1,12,1,1,78,78,1,1,364,2366,364,1,1,1365,41405,41405,1365,1,1,4368,496860,2318680,496860,4368,1,1,12376,4504864,78835120,78835120,4504864,12376,1,1,31824,32821152,1837984512,6892441920,1837984512,32821152,31824,1,
+A342891 ,1,1,1,1,13,1,1,91,91,1,1,455,3185,455,1,1,1820,63700,63700,1820,1,1,6188,866320,4331600,866320,6188,1,1,18564,8836464,176729280,176729280,8836464,18564,1,1,50388,71954064,4892876352,19571505408,4892876352,71954064,50388,1,
+A342892 ,1,1,1,1,1,0,1,0,1,1,0,1,1,0,0,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0,1,1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0,0,1,1,1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,1,1,0,0,1,0,1,1,0,1,0,
+A342893 ,54,5104,811538,
+A342894 ,28,1225,16578,1479604544,1885800643779,
+A342895 ,29,766,4225,13124675,224688658,
+A342896 ,30,221,2676,696467,17886882,
+A342897 ,129,13896,5978882,
+A342898 ,52,1979,236674,
+A342899 ,53,1252,20995,287618651,
+A342900 ,36,626,6626,12047994,353563195,
+A342901 ,37,470,2932,893604,82718691,
+A342902 ,1729,251,219,157,158,131,132,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,
+A342903 ,50,27,28,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,
+A342904 ,5,27,251,2673,1375298099,106426514,
+A342905 ,1,2,2,3,2,3,2,6,6,2,5,2,3,2,5,6,10,6,6,10,6,7,6,15,2,15,6,7,2,14,6,10,10,6,14,2,3,2,21,6,5,6,21,2,3,10,6,6,14,30,30,14,6,6,10,11,10,3,2,35,6,35,2,3,10,11,6,22,30,6,10,42,42,10,6,30,22,6,
+A342906 ,0,2,11,50,214,892,3667,14954,60674,245348,989790,3986292,16034316,64434424,258740611,1038384154,4165322506,16702230484,66952213546,268313786524,1075045360756,4306563947464,17249126430766,69078840030340,276613030309204,1107532553770472,4434066076454492,
+A342907 ,1,4,20,304,6784,407684,39072966,9449433606,3830070645700,3762885306351756,6402694828334379856,
A342908 ,1,1,1,1,1,2,2,1,3,5,3,4,1,4,9,9,4,12,10,12,1,5,14,19,14,5,25,35,42,18,35,25,34,1,6,20,34,34,20,6,44,84,100,72,140,100,28,72,84,44,136,112,112,136,1,7,27,55,69,55,27,7,70,168,198,196,378,268,126,324,378,198,40,126,196,168,70,364,504,504,612,256,420,504,256,504,364,496,
+A342909 ,1,0,1,2,0,1,4,0,2,7,0,3,2,9,0,4,13,0,5,3,13,20,0,6,5,18,1,10,4,18,28,0,7,24,1,12,7,26,1,13,33,0,8,13,36,0,9,32,2,19,7,31,3,22,7,33,54,0,
A342911 ,1,0,1,0,1,4,0,1,8,15,0,1,16,35,54,0,1,32,83,134,185,0,1,64,199,340,481,622,0,1,128,479,872,1265,1658,2051,0,1,256,1155,2254,3361,4468,5575,6682,0,1,512,2787,5854,8993,12132,15271,18410,21549,
+A342912 ,1,1,3,6,15,36,91,232,603,1585,4213,11298,30537,83097,227475,625992,1730787,4805595,13393689,37458330,105089229,295673994,834086421,2358641376,6684761125,18985057351,54022715451,154000562758,439742222071,1257643249140,3602118427251,
A342913 ,1,1,3,1,2,2,5,1,4,2,3,3,7,1,6,2,5,3,4,4,9,1,8,2,7,3,6,4,5,5,11,1,10,2,9,3,8,4,7,5,6,6,13,1,12,2,11,3,10,4,9,5,8,6,7,7,15,1,14,2,13,3,12,4,11,5,10,6,9,7,8,8,17,1,16,2,15,3,14,4,13,5,12,6,11,7,
+A342914 ,1,12,36,73,123,186,262,351,453,568,696,837,991,1158,1338,1531,1737,1956,2188,2433,2691,2962,3246,3543,3853,4176,4512,4861,5223,5598,5986,6387,6801,7228,7668,8121,8587,9066,9558,10063,10581,11112,11656,12213,12783,13366,
A342915 ,1,3,4,1,6,1,8,3,2,1,12,1,14,3,8,1,18,1,20,3,2,1,24,1,2,3,4,1,30,1,32,3,2,1,12,1,38,3,8,1,42,1,44,9,2,1,48,1,2,3,4,1,54,1,8,3,2,1,60,1,62,3,32,1,6,1,68,3,2,1,72,1,74,3,4,1,6,1,80,9,2,1,84,1,2,3,8,1,90,1,4,3,2,1,24,1,98,3,4,1,102,
A342916 ,2,1,1,5,1,7,1,3,5,11,1,13,1,5,2,17,1,19,1,7,11,23,1,25,13,9,7,29,1,31,1,11,17,35,3,37,1,13,5,41,1,43,1,5,23,47,1,49,25,17,13,53,1,55,7,19,29,59,1,61,1,21,2,65,11,67,1,23,35,71,1,73,1,25,19,77,13,79,1,9,41,83,1,85,43,29,11,89,1,91,23,31,
A342917 ,1,1,1,6,1,12,1,4,6,18,1,24,1,8,3,24,1,36,1,12,16,36,1,48,15,14,9,48,1,72,1,16,24,54,4,72,1,20,7,72,1,96,1,8,36,72,1,96,28,30,18,84,1,108,9,32,40,90,1,144,1,32,3,96,14,144,1,36,48,144,1,144,1,38,30,120,16,168,1,16,54,126,1,192,54,44,15,144,
A342918 ,1,1,1,5,1,7,1,3,5,11,1,13,1,5,2,17,1,19,1,7,11,23,1,25,13,9,7,29,1,31,1,11,17,35,3,37,1,13,5,41,1,43,1,15,23,47,1,49,25,17,13,53,1,55,7,19,29,59,1,61,1,21,8,65,11,67,1,23,35,71,1,73,1,25,19,77,13,79,1,27,41,83,1,85,43,29,11,89,1,91,23,
A342919 ,0,1,1,2,1,5,1,1,1,7,1,2,1,3,1,4,1,7,1,2,5,13,1,11,1,5,3,2,1,31,1,5,7,19,1,5,1,7,2,17,1,41,1,2,13,25,1,7,1,1,5,2,1,3,2,23,11,31,1,23,1,11,17,2,3,61,1,2,13,59,1,13,1,13,11,2,3,71,1,11,1,43,1,31,11,15,4,35,1,41,5,2,17,49,1,17,1,11,25,
+A342920 ,1,1,1,2,1,2,1,8,12,2,1,4,1,2,6,16,1,24,1,4,6,2,1,26,50,2,16,4,1,62,1,10,6,2,126,48,1,2,6,18,1,24,1,4,46,2,1,22,1486,100,6,4,1,32,94,8,6,2,1,54,1,2,72,20,264,12,1,4,6,120,1,376,1,2,1142,4,242,12,1,36,342,2,1,48,272,2,6,8,1,92,318,
+A342921 ,0,1,1,5,1,7,8,31,1,9,10,41,12,59,71,247,1,13,14,61,16,87,103,371,18,113,131,493,167,719,886,2927,1,15,16,71,18,101,119,433,20,131,151,575,191,837,1028,3421,24,191,215,859,263,1241,1504,5153,311,1623,1934,6871,2556,10117,12673,40361,1,19,20,91,22,129,
+A342922 ,6,28,29,496,857,1721,8128,164284,6511664,33550336,
+A342923 ,120,672,963,1036,264768,523776,459818240,
+A342924 ,6,28,120,496,672,963,1036,5871,8128,10479,164284,264768,523776,2308203,6511664,33550336,41240261,75384301,400902412,459818240,
+A342925 ,0,1,4,1,5,16,12,8,1,21,16,32,9,44,44,1,21,16,24,41,80,60,44,92,1,41,68,92,31,156,80,51,112,81,112,20,21,92,92,123,41,272,48,124,71,156,112,128,22,34,156,77,81,244,156,244,176,123,92,332,33,272,164,1,124,384,72,165,272,384,156,119,39,101,128,188,
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+A343501 ,4,6,14,16,20,22,24,30,36,38,46,52,54,56,62,64,68,70,78,80,84,86,88,94,96,100,102,110,116,118,120,126,132,134,142,144,148,150,152,158,164,166,174,180,182,184,190,196,198,206,208,212,214,216,222,224,228,
+A343503 ,1,2,2,3,4,6,5,5,6,4,4,5,6,4,4,8,9,6,9,8,8,6,8,7,2,7,6,6,5,7,9,8,7,10,6,11,9,9,10,6,10,9,10,6,7,10,10,6,7,6,7,7,6,7,6,11,10,9,9,9,10,10,10,9,7,7,14,8,11,9,13,11,7,13,9,7,10,8,6,7,10,11,4,9,8,12,8,11,12,6,12,11,12,13,7,12,10,11,11,9,
+A343504 ,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,6,6,1,1,1,1,2,2,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,6,6,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,6,6,6,6,6,6,3,3,3,3,6,6,3,3,3,3,6,6,6,6,6,
+A343505 ,1,1,2,2,4,1,6,3,6,2,60,2,120,3,3,6,1008,4,51480,1,4,30,6930,1,140,36,60,20,16380,4,243374040,12,105,504,12,6,6126120,4680,168,3,314954640,10,209969760,24,4,180180,1790848659600,6,924,6,660,1260,8303710615200,
+A343508 ,1,65,731,4162,15629,47515,117655,266372,532905,1015885,1771571,3042422,4826821,7647575,11424799,17047816,24137585,34638825,47045899,65047898,86005805,115152115,148035911,194717932,244203145,313743365,388487763,489680110,594823349,
+A343509 ,1,129,2189,16514,78129,282381,823549,2113796,4787349,10078641,19487181,36149146,62748529,106237821,171024381,270565896,410338689,617568021,893871757,1290222306,1802748761,2513846349,3404825469,4627099444,6103828145,8094560241,
+A343510 ,1,1,3,1,5,5,1,9,11,8,1,17,29,22,9,1,33,83,74,29,15,1,65,245,274,129,55,13,1,129,731,1058,629,261,55,20,1,257,2189,4162,3129,1411,349,92,21,1,513,6563,16514,15629,8085,2407,596,105,27,1,1025,19685,65794,78129,47515,16813,4388,789,145,21,
+A343511 ,1,2,2,6,2,10,2,42,6,10,2,146,2,10,10,1806,2,146,2,146,10,10,2,23226,6,10,42,146,2,314,2,3263442,10,10,10,42814,2,10,10,23226,2,314,2,146,146,10,2,542731938,6,146,10,146,2,23226,10,23226,10,10,2,141578,2,10,146,10650056950806,10,
+A343512 ,1,6,28,72,90,92,96,112,118,148,160,162,184,222,282,312,314,316,330,336,390,396,418,440,444,448,472,488,524,534,552,598,604,614,638,748,758,798,824,848,906,916,970,992,1008,1010,1012,1016,1056,1078,1084,1094,1098,
+A343513 ,1,2,10,30,101,137,442,526,1063,1202,3026,1965,6085,4853,7310,8654,18497,10100,29242,17630,29557,30857,64010,30397,77601,60842,89272,71913,164837,60737,216226,139470,188165,180338,265142,152544,443557,282665,371134,275726,672401,251066,815410,461645,
+A343514 ,1,2,18,84,355,645,2276,3192,7413,9400,25334,18395,60711,52747,88760,106688,243849,137790,432346,275570,499867,522513,1151404,561415,1542125,1214436,1907502,1569673,3756719,1344999,5274000,3451216,4970577,4690778,7499154,4217504,12948595,8207261,11565572,
+A343516 ,1,1,3,1,4,5,1,5,8,8,1,6,12,15,9,1,7,17,26,19,15,1,8,23,42,39,35,13,1,9,30,64,74,76,34,20,1,10,38,93,130,153,90,56,21,1,11,47,130,214,287,216,152,63,27,1,12,57,176,334,506,468,379,191,86,21,
+A343517 ,1,4,12,42,130,506,1722,6622,24426,93427,352726,1359388,5200312,20097156,77567064,300787366,1166803126,4539197723,17672631918,68933307843,269129530770,1052113994340,4116715363822,16124224571368,63205303313900,247961973949536,
+A343518 ,1,6,17,42,74,153,216,379,531,809,1011,1605,1832,2626,3268,4304,4861,6798,7333,9878,11148,13711,14972,19985,20775,25643,28503,34517,35988,46162,46406,57092,61077,70986,75099,92520,91426,108693,115774,135491,135791,165719,163227,193437,
+A343519 ,1,7,23,64,130,287,468,864,1335,2156,3013,4790,6200,9072,11972,16440,20365,28209,33667,45014,54192,68853,80752,104964,119279,148778,172629,211252,237364,295288,324662,394368,442133,522403,578385,696624,749434,884443,975250,1136476,
+A343520 ,1,8,30,93,214,506,930,1818,3065,5247,8018,13080,18576,28104,39300,56184,74629,104978,134614,182897,232258,304098,376762,492068,594635,754941,912384,1137106,1344932,1674374,1947822,2382888,2776997,3337364,3843360,4629687,5245822,6231194,
+A343521 ,1,9,38,130,334,846,1722,3572,6513,11806,19458,32948,50400,79290,117092,174256,245173,354249,480718,670420,891690,1203578,1560802,2076496,2630915,3416352,4285152,5461348,6724548,8490884,10295502,12798224,15420213,18888861,
+A343528 ,0,1,3,4,5,5,3,4,6,5,5,6,5,6,6,4,7,10,10,9,7,4,7,10,7,8,9,7,5,7,7,10,13,9,8,7,5,8,14,9,10,11,6,9,10,8,10,13,8,7,6,5,11,15,9,7,8,6,8,10,10,10,10,6,7,9,6,10,17,10,9,9,6,10,10,6,9,9,6,10,9,6,11,14,8,11,11,9,11,11,
+A343531 ,2,7,15,31,51,83,119,171,231,307,395,503,627,755,919,1079,1271,1483,1703,1967,2215,2495,2795,3127,3479,3839,4267,4647,5059,5539,5991,6511,7063,7651,8211,8855,9439,10139,10887,11611,12371,13159,13951,14715,15647,16591,17431,18487,19419,20415,21491,
+A343532 ,2,7,31,83,307,503,919,1483,5059,9439,10139,13159,15647,17431,21491,23671,30911,33599,47459,49199,52627,58199,62327,79379,81551,90971,98443,109171,114643,123439,162007,168863,172331,175811,278767,298303,303011,322951,376399,387631,393007,571531,592531,
+A343538 ,13,61,63,67,19,99,31,69,91,87,79,37,39,43,27,49,73,51,
diff --git a/demos/IntegerTriangles-checkpoint.ipynb b/demos/IntegerTriangles-checkpoint.ipynb
new file mode 100644
index 0000000..4345a97
--- /dev/null
+++ b/demos/IntegerTriangles-checkpoint.ipynb
@@ -0,0 +1,1479 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Integer Triangles Trait Cards
Computed with Julia
(See also the Tutorial)
"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Pkg.add(PackageSpec(url=\"https://github.com/OpenLibMathSeq/IntegerTriangles.jl\"))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "Welcome to Nemo version 0.22.0\n",
+ "\n",
+ "Nemo comes with absolutely no warranty whatsoever\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "using Nemo;\n",
+ "using IntegerTriangles;"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Definition: A *integer triangle* is an array of arrays whose members are integers. \n",
+ "It has the type **ℤTri**.\n",
+ "\n",
+ "The row of an integer triangle is an integer sequence, it has the type **ℤSeq**.\n",
+ "\n",
+ "An integer is a multiple precision integer which is created by the constructor **ZZ**.\n",
+ "\n",
+ "Examples for the creation of an integer triangle:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "7-element Vector{Vector{fmpz}}:\n",
+ " [0]\n",
+ " [0, 1]\n",
+ " [0, 1, 2]\n",
+ " [0, 1, 2, 3]\n",
+ " [0, 1, 2, 3, 4]\n",
+ " [0, 1, 2, 3, 4, 5]\n",
+ " [0, 1, 2, 3, 4, 5, 6]"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "[[ZZ(k) for k in 0:n] for n in 0:6]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "7-element Vector{fmpz}:\n",
+ " 0\n",
+ " 1\n",
+ " 4\n",
+ " 9\n",
+ " 16\n",
+ " 25\n",
+ " 36"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "ZSeq(7, n -> n*n)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[0]\n",
+ "[0, 1]\n",
+ "[0, 1, 2]\n",
+ "[0, 1, 2, 3]\n",
+ "[0, 1, 2, 3, 4]\n",
+ "[0, 1, 2, 3, 4, 5]\n"
+ ]
+ }
+ ],
+ "source": [
+ "T = Telescope(6, n -> ZZ(n)) \n",
+ "Show(collect(T))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The shape of the triangle is /not/ fixed. It includes the cases denoted by the OEIS keywords *tabl* and *tabf*. We allow the empty sequence to be element of an integer triangle."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Divisors (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function Divisors(n) \n",
+ " n == 0 && return ℤSeq[]\n",
+ " (ZZ(d) for d in 1:n if rem(n, d) == 0)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "true\n"
+ ]
+ }
+ ],
+ "source": [
+ "dk = collect(Divisors(12))\n",
+ "println(isa(dk, ℤSeq))\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "sum (generic function with 17 methods)"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "import Base.sum\n",
+ "sum(T::ℤTri) = sum.(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "$8$"
+ ],
+ "text/latex": [
+ "8"
+ ],
+ "text/plain": [
+ "8"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "sum(Divisors(7))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "5-element Vector{fmpz}:\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "S = ZSeq(5) "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "5-element Vector{Vector{fmpz}}:\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef\n",
+ " #undef"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T = ZTri(5) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If the keyword 'reg' is set to true, the constructor returns an uninitialized regular triangle."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6-element Vector{Vector{fmpz}}:\n",
+ " [#undef]\n",
+ " [#undef, #undef]\n",
+ " [#undef, #undef, #undef]\n",
+ " [#undef, #undef, #undef, #undef]\n",
+ " [#undef, #undef, #undef, #undef, #undef]\n",
+ " [#undef, #undef, #undef, #undef, #undef, #undef]"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T = ZTri(6, reg=true) "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6-element Vector{Vector{fmpz}}:\n",
+ " [1]\n",
+ " [1, 2]\n",
+ " [1, 2, 3]\n",
+ " [1, 2, 3, 4]\n",
+ " [1, 2, 3, 4, 5]\n",
+ " [1, 2, 3, 4, 5, 6]"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "for n in 1:6 T[n] = [ZZ(k) for k in 1:n] end\n",
+ "T"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "3-element Vector{fmpz}:\n",
+ " 1\n",
+ " 2\n",
+ " 3"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T[3]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "$2$"
+ ],
+ "text/latex": [
+ "2"
+ ],
+ "text/plain": [
+ "2"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T[3][2]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "true\n",
+ "true\n",
+ "true\n"
+ ]
+ }
+ ],
+ "source": [
+ "isa(T, ℤTri) |> println\n",
+ "isa(T[3], ℤSeq) |> println\n",
+ "isa(T[3][2], ℤInt) |> println"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "7-element Vector{Vector{fmpz}}:\n",
+ " [1]\n",
+ " [0, 1]\n",
+ " [0, 2, 1]\n",
+ " [0, 6, 6, 1]\n",
+ " [0, 24, 36, 12, 1]\n",
+ " [0, 120, 240, 120, 20, 1]\n",
+ " [0, 720, 1800, 1200, 300, 30, 1]"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function LahIndexed(n, k)\n",
+ " function recLah(n, k)\n",
+ " k < 0 && return ZZ(0)\n",
+ " k == n && return ZZ(1)\n",
+ " recLah(n-1, k-1) + recLah(n-1, k)*(n+k-1)\n",
+ " end\n",
+ " recLah(n, k)\n",
+ "end\n",
+ "\n",
+ "[[LahIndexed(n, k) for k in 0:n] for n in 0:6]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Thus LahNumbers(n) returns an n-element Array{Array{ℤInt,1},1}\n",
+ "which is according to our definiton a triangle with *n* rows."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "LahNumbers (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "const cacheLah = Dict{Int, ℤSeq}([0 => [ZZ(1)]])\n",
+ "\n",
+ "function LahNumbers(n)\n",
+ " haskey(cacheLah, n) && return cacheLah[n]\n",
+ " prevrow = LahNumbers(n-1)\n",
+ " row = ZSeq(n+1)\n",
+ " row[1] = 0; row[n+1] = 1\n",
+ " for k in 2:n\n",
+ " row[k] = prevrow[k-1] + prevrow[k]*(n+k-2) \n",
+ " end\n",
+ " cacheLah[n] = row\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6-element Vector{fmpz}:\n",
+ " 0\n",
+ " 120\n",
+ " 240\n",
+ " 120\n",
+ " 20\n",
+ " 1"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "LahNumbers(5)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Dict{Int64, Vector{fmpz}}(0 => [1], 4 => [0, 24, 36, 12, 1], 5 => [0, 120, 240, 120, 20, 1], 2 => [0, 2, 1], 3 => [0, 6, 6, 1], 1 => [0, 1])\n"
+ ]
+ }
+ ],
+ "source": [
+ "println(cacheLah)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let's check the time and space consumtion: "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 0.002507 seconds (10.17 k allocations: 212.609 KiB)\n"
+ ]
+ }
+ ],
+ "source": [
+ "@time LahNumbers(100);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "$1200$"
+ ],
+ "text/latex": [
+ "1200"
+ ],
+ "text/plain": [
+ "1200"
+ ]
+ },
+ "execution_count": 22,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "LahNumbers(n, k) = LahNumbers(n+1)[k+1]\n",
+ "LahNumbers(5, 3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "# 2 methods for generic function LahNumbers:- LahNumbers(n) in Main at In[18]:3
- LahNumbers(n, k) in Main at In[22]:1
"
+ ],
+ "text/plain": [
+ "# 2 methods for generic function \"LahNumbers\":\n",
+ "[1] LahNumbers(n) in Main at In[18]:3\n",
+ "[2] LahNumbers(n, k) in Main at In[22]:1"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "methods(LahNumbers)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "LahTriangle (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function LahTriangle(size) \n",
+ " length(cacheLah) < size && LahNumbers(size)\n",
+ " [cacheLah[n] for n in 0:size-1] \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "# 1 method for generic function LahTriangle:- LahTriangle(size) in Main at In[24]:1
"
+ ],
+ "text/plain": [
+ "# 1 method for generic function \"LahTriangle\":\n",
+ "[1] LahTriangle(size) in Main at In[24]:1"
+ ]
+ },
+ "execution_count": 25,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "methods(LahTriangle)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "7"
+ ]
+ },
+ "execution_count": 26,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T = LahTriangle(7)\n",
+ "length(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "altsum (generic function with 2 methods)"
+ ]
+ },
+ "execution_count": 27,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "evensum(A) = sum(A[1:2:end]) \n",
+ "oddsum(A) = sum(A[2:2:end])\n",
+ "altsum(A) = evensum(A) - oddsum(A)\n",
+ "evensum(T::ℤTri) = evensum.(T)\n",
+ "oddsum(T::ℤTri) = oddsum.(T)\n",
+ "altsum(T::ℤTri) = evensum(T) - oddsum(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 28,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]\n",
+ "[1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536]\n",
+ "[0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017]\n",
+ "[1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]\n"
+ ]
+ }
+ ],
+ "source": [
+ "T = LahTriangle(10)\n",
+ "Println.([sum(T), evensum(T), oddsum(T), altsum(T)]);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1, 0, 2, 6, 36, 240, 1200, 12600, 58800, 846720]\n"
+ ]
+ }
+ ],
+ "source": [
+ "middle(A) = A[div(end + 1, 2)] \n",
+ "middle(T::ℤTri) = middle.(T)\n",
+ "\n",
+ "middle(LahTriangle(10)) |> Println"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1, 2, 36, 1200, 58800, 3810240, 307359360, 29682132480]\n"
+ ]
+ }
+ ],
+ "source": [
+ "central(T::ℤTri) = middle.(T[1:2:end])\n",
+ "\n",
+ "central(LahTriangle(16)) |> Println"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "DiagonalTriangle (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function DiagonalTriangle(T::ℤTri)\n",
+ " dim = length(T)\n",
+ " U = ZTri(dim)\n",
+ " for n in 1:dim\n",
+ " R = ZSeq(div(n+1,2))\n",
+ " for k in 0:div(n-1,2)\n",
+ " R[k+1] = T[n-k][k+1]\n",
+ " end\n",
+ " U[n] = R\n",
+ " end\n",
+ " U\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "10-element Vector{Vector{fmpz}}:\n",
+ " [1]\n",
+ " [0]\n",
+ " [0, 1]\n",
+ " [0, 2]\n",
+ " [0, 6, 1]\n",
+ " [0, 24, 6]\n",
+ " [0, 120, 36, 1]\n",
+ " [0, 720, 240, 12]\n",
+ " [0, 5040, 1800, 120, 1]\n",
+ " [0, 40320, 15120, 1200, 20]"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "T = LahTriangle(10) \n",
+ "DiagonalTriangle(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "9-element Vector{fmpz}:\n",
+ " 1\n",
+ " 0\n",
+ " 1\n",
+ " 2\n",
+ " 7\n",
+ " 30\n",
+ " 157\n",
+ " 972\n",
+ " 6961"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "diagsum(T) = sum(DiagonalTriangle(T))\n",
+ "diagsum(LahTriangle(9))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 34,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "leftside (generic function with 2 methods)"
+ ]
+ },
+ "execution_count": 34,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "leftside(A) = A[1] \n",
+ "rightside(A) = A[end] \n",
+ "\n",
+ "rightside(T::ℤTri) = rightside.(T)\n",
+ "leftside( T::ℤTri) = leftside.(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "profile (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function profile(T::ℤTri)\n",
+ " println(\"Triangle: \");\n",
+ " for row in T Println(row) end; println()\n",
+ " print(\"Sum: \"); sum(T) |> Println \n",
+ " print(\"EvenSum: \"); evensum(T) |> Println \n",
+ " print(\"OddSum: \"); oddsum(T) |> Println \n",
+ " print(\"AltSum: \"); altsum(T) |> Println \n",
+ " print(\"DiagSum: \"); diagsum(T) |> Println \n",
+ " print(\"Middle: \"); middle(T) |> Println \n",
+ " print(\"Central: \"); central(T) |> Println \n",
+ " print(\"LeftSide: \"); leftside(T) |> Println \n",
+ " print(\"RightSide:\"); rightside(T) |> Println \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 36,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Triangle: \n",
+ "[1]\n",
+ "[0, 1]\n",
+ "[0, 2, 1]\n",
+ "[0, 6, 6, 1]\n",
+ "[0, 24, 36, 12, 1]\n",
+ "[0, 120, 240, 120, 20, 1]\n",
+ "[0, 720, 1800, 1200, 300, 30, 1]\n",
+ "[0, 5040, 15120, 12600, 4200, 630, 42, 1]\n",
+ "[0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1]\n",
+ "[0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1]\n",
+ "\n",
+ "Sum: [1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553]\n",
+ "EvenSum: [1, 0, 1, 6, 37, 260, 2101, 19362, 201097, 2326536]\n",
+ "OddSum: [0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017]\n",
+ "AltSum: [1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519]\n",
+ "DiagSum: [1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660]\n",
+ "Middle: [1, 0, 2, 6, 36, 240, 1200, 12600, 58800, 846720]\n",
+ "Central: [1, 2, 36, 1200, 58800]\n",
+ "LeftSide: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]\n",
+ "RightSide:[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]\n"
+ ]
+ }
+ ],
+ "source": [
+ "profile(LahTriangle(10))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "true\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "10-element Vector{Vector{fmpz}}:\n",
+ " [1]\n",
+ " [0, 1]\n",
+ " [0, -2, 1]\n",
+ " [0, 6, -6, 1]\n",
+ " [0, -24, 36, -12, 1]\n",
+ " [0, 120, -240, 120, -20, 1]\n",
+ " [0, -720, 1800, -1200, 300, -30, 1]\n",
+ " [0, 5040, -15120, 12600, -4200, 630, -42, 1]\n",
+ " [0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1]\n",
+ " [0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1]"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "invT = InverseTriangle(T) \n",
+ "isa(invT, ℤTri) |> println\n",
+ "invT"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 38,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1, 6, 84, 1680, 39240, 999216, 26899896, 752939424]\n"
+ ]
+ }
+ ],
+ "source": [
+ "Apery = ℤInt[\n",
+ " 1,\n",
+ " 5,\n",
+ " 73,\n",
+ " 1445,\n",
+ " 33001,\n",
+ " 819005,\n",
+ " 21460825,\n",
+ " 584307365\n",
+ " ]\n",
+ "\n",
+ "BinomialTransform(Apery) |> Println"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1]\n",
+ "[0, 1]\n",
+ "[0, 1, 1]\n",
+ "[0, 1, 3, 1]\n",
+ "[0, 1, 6, 6, 1]\n",
+ "[0, 1, 10, 20, 10, 1]\n",
+ "[0, 1, 15, 50, 50, 15, 1]\n",
+ "[0, 1, 21, 105, 175, 105, 21, 1]\n",
+ "[0, 1, 28, 196, 490, 490, 196, 28, 1]\n"
+ ]
+ }
+ ],
+ "source": [
+ "T = NarayanaTriangle(8)\n",
+ "Show(T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "true"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "isa(T, ℤTri) "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 41,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "=================\n",
+ "NarayanaTriangle \n",
+ "\n",
+ "Triangle: [1, 0, 1, 0, 1, 1, 0, 1]\n",
+ "Sum: [1, 1, 2, 5, 14, 42, 132, 429]\n",
+ "EvenSum: [1, 0, 1, 3, 7, 20, 66, 217]\n",
+ "OddSum: [0, 1, 1, 2, 7, 22, 66, 212]\n",
+ "AltSum: [1, -1, 0, 1, 0, -2, 0, 5]\n",
+ "DiagSum: [1, 0, 1, 1, 2, 4, 8, 17]\n",
+ "Central: [1, 1, 6, 50, 490]\n",
+ "LeftSide: [1, 0, 0, 0, 0, 0, 0, 0]\n",
+ "RightSide: [1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "PosHalf: [1, 1, 3, 11, 45, 197, 903, 4279]\n",
+ "NegHalf: [1, 1, -1, -1, 5, -3, -21, 51]\n",
+ "PolyVal2: [511, 3586, 29692, 275978, 2788324, 29938732, 336347712, 3911669402]\n",
+ "PolyVal3: [9841, 73812, 634776, 6059904, 62503368, 682458804, 7775247636, 91507441320]\n",
+ "N0TS: [0, 1, 3, 10, 35, 126, 462, 1716]\n",
+ "NATS: [1, 2, 5, 15, 49, 168, 594, 2145]\n",
+ "\n",
+ "\n",
+ "Triangle:\n",
+ "\n",
+ "[1]\n",
+ "[0, 1]\n",
+ "[0, 1, 1]\n",
+ "[0, 1, 3, 1]\n",
+ "[0, 1, 6, 6, 1]\n",
+ "[0, 1, 10, 20, 10, 1]\n",
+ "[0, 1, 15, 50, 50, 15, 1]\n",
+ "[0, 1, 21, 105, 175, 105, 21, 1]\n",
+ "\n",
+ "Flat triangle: [1, 0, 1, 0, 1, 1, 0, 1]\n",
+ "\n",
+ "Inverse triangle:\n",
+ "\n",
+ "[1]\n",
+ "[0, 1]\n",
+ "[0, -1, 1]\n",
+ "[0, 2, -3, 1]\n",
+ "[0, -7, 12, -6, 1]\n",
+ "[0, 39, -70, 40, -10, 1]\n",
+ "[0, -321, 585, -350, 100, -15, 1]\n",
+ "[0, 3681, -6741, 4095, -1225, 210, -21, 1]\n",
+ "\n",
+ "Inverse: [1, 0, 1, 0, -1, 1, 0, 2]\n",
+ "\n",
+ "Diagonal triangle:\n",
+ "\n",
+ "[1]\n",
+ "[0]\n",
+ "[0, 1]\n",
+ "[0, 1]\n",
+ "[0, 1, 1]\n",
+ "[0, 1, 3]\n",
+ "[0, 1, 6, 1]\n",
+ "[0, 1, 10, 6]\n",
+ "\n",
+ "Diagonal: [1, 0, 0, 1, 0, 1, 0, 1]\n",
+ "\n",
+ "Polynomial values, array:\n",
+ "\n",
+ "[1, 1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "[0, 1, 2, 3, 4, 5, 6, 7, 8]\n",
+ "[0, 2, 6, 12, 20, 30, 42, 56, 72]\n",
+ "[0, 5, 22, 57, 116, 205, 330, 497, 712]\n",
+ "[0, 14, 90, 300, 740, 1530, 2814, 4760, 7560]\n",
+ "[0, 42, 394, 1686, 5028, 12130, 25422, 48174, 84616]\n",
+ "[0, 132, 1806, 9912, 35700, 100380, 239442, 507696, 985032]\n",
+ "[0, 429, 8558, 60213, 261780, 857405, 2326434, 5516133, 11814728]\n",
+ "[0, 1430, 41586, 374988, 1967300, 7503330, 23151030, 61363736, 145043208]\n",
+ "\n",
+ "Polynomial values, triangle:\n",
+ "\n",
+ "[1]\n",
+ "[0, 1]\n",
+ "[0, 1, 1]\n",
+ "[0, 2, 2, 1]\n",
+ "[0, 5, 6, 3, 1]\n",
+ "[0, 14, 22, 12, 4, 1]\n",
+ "[0, 42, 90, 57, 20, 5, 1]\n",
+ "[0, 132, 394, 300, 116, 30, 6, 1]\n",
+ "\n",
+ "PolyVal: [1, 0, 1, 0, 1, 1, 0, 2]\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "dim = 8\n",
+ "TraitCard(NarayanaTriangle, dim)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 42,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "=================\n",
+ "LaguerreTriangle \n",
+ "\n",
+ "Triangle: [1, 1, 1, 2, 4, 1, 6, 18]\n",
+ "Sum: [1, 2, 7, 34, 209, 1546, 13327, 130922]\n",
+ "EvenSum: [1, 1, 3, 15, 97, 745, 6571, 65359]\n",
+ "OddSum: [0, 1, 4, 19, 112, 801, 6756, 65563]\n",
+ "AltSum: [1, 0, -1, -4, -15, -56, -185, -204]\n",
+ "DiagSum: [1, 1, 3, 10, 43, 225, 1393, 9976]\n",
+ "Central: [1, 4, 72, 2400]\n",
+ "LeftSide: [1, 1, 2, 6, 24, 120, 720, 5040]\n",
+ "RightSide: [1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "PosHalf: [1, 3, 17, 139, 1473, 19091, 291793, 5129307]\n",
+ "NegHalf: [1, -1, 1, 7, -127, 1711, -23231, 334391]\n",
+ "PolyVal2: [255, 1793, 16384, 179714, 2276866, 32531468, 515561990, 8952335236]\n",
+ "PolyVal3: [3280, 24604, 232916, 2617020, 33756348, 489160284, 7841902404, 137482567476]\n",
+ "N0TS: [0, 1, 6, 39, 292, 2505, 24306, 263431]\n",
+ "NATS: [1, 3, 13, 73, 501, 4051, 37633, 394353]\n",
+ "\n",
+ "\n",
+ "Triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[2, 4, 1]\n",
+ "[6, 18, 9, 1]\n",
+ "[24, 96, 72, 16, 1]\n",
+ "[120, 600, 600, 200, 25, 1]\n",
+ "[720, 4320, 5400, 2400, 450, 36, 1]\n",
+ "[5040, 35280, 52920, 29400, 7350, 882, 49, 1]\n",
+ "\n",
+ "Flat triangle: [1, 1, 1, 2, 4, 1, 6, 18]\n",
+ "\n",
+ "Inverse triangle:\n",
+ "\n",
+ "[1]\n",
+ "[-1, 1]\n",
+ "[2, -4, 1]\n",
+ "[-6, 18, -9, 1]\n",
+ "[24, -96, 72, -16, 1]\n",
+ "[-120, 600, -600, 200, -25, 1]\n",
+ "[720, -4320, 5400, -2400, 450, -36, 1]\n",
+ "[-5040, 35280, -52920, 29400, -7350, 882, -49, 1]\n",
+ "\n",
+ "Inverse: [1, -1, 1, 2, -4, 1, -6, 18]\n",
+ "\n",
+ "Diagonal triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1]\n",
+ "[2, 1]\n",
+ "[6, 4]\n",
+ "[24, 18, 1]\n",
+ "[120, 96, 9]\n",
+ "[720, 600, 72, 1]\n",
+ "[5040, 4320, 600, 16]\n",
+ "\n",
+ "Diagonal: [1, 1, 2, 1, 6, 4, 24, 18]\n",
+ "\n",
+ "Polynomial values, array:\n",
+ "\n",
+ "[1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "[1, 2, 3, 4, 5, 6, 7, 8]\n",
+ "[2, 7, 14, 23, 34, 47, 62, 79]\n",
+ "[6, 34, 86, 168, 286, 446, 654, 916]\n",
+ "[24, 209, 648, 1473, 2840, 4929, 7944, 12113]\n",
+ "[120, 1546, 5752, 14988, 32344, 61870, 108696, 179152]\n",
+ "[720, 13327, 58576, 173007, 414160, 866695, 1649232, 2921911]\n",
+ "[5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, 51988748]\n",
+ "\n",
+ "Polynomial values, triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[2, 2, 1]\n",
+ "[6, 7, 3, 1]\n",
+ "[24, 34, 14, 4, 1]\n",
+ "[120, 209, 86, 23, 5, 1]\n",
+ "[720, 1546, 648, 168, 34, 6, 1]\n",
+ "[5040, 13327, 5752, 1473, 286, 47, 7, 1]\n",
+ "\n",
+ "PolyVal: [1, 1, 1, 2, 2, 1, 6, 7]\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "TraitCard(LaguerreTriangle, dim)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 43,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "=================\n",
+ "MotzkinTriangle \n",
+ "\n",
+ "Triangle: [1, 1, 1, 2, 2, 1, 4, 5]\n",
+ "Sum: [1, 2, 5, 13, 35, 96, 267, 750]\n",
+ "EvenSum: [1, 1, 3, 7, 19, 51, 141, 393]\n",
+ "OddSum: [0, 1, 2, 6, 16, 45, 126, 357]\n",
+ "AltSum: [1, 0, 1, 1, 3, 6, 15, 36]\n",
+ "DiagSum: [1, 1, 3, 6, 15, 36, 91, 232]\n",
+ "Central: [1, 2, 9, 44]\n",
+ "LeftSide: [1, 1, 2, 4, 9, 21, 51, 127]\n",
+ "RightSide: [1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "PosHalf: [1, 3, 13, 59, 285, 1419, 7245, 37659]\n",
+ "NegHalf: [1, -1, 5, -17, 77, -345, 1653, -8097]\n",
+ "PolyVal2: [255, 1793, 13308, 100878, 775339, 6019401, 47095629, 370752919]\n",
+ "PolyVal3: [3280, 24604, 190268, 1486888, 11701536, 92579016, 735578160, 5864923504]\n",
+ "N0TS: [0, 1, 4, 14, 46, 147, 462, 1437]\n",
+ "NATS: [1, 3, 9, 27, 81, 243, 729, 2187]\n",
+ "\n",
+ "\n",
+ "Triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[2, 2, 1]\n",
+ "[4, 5, 3, 1]\n",
+ "[9, 12, 9, 4, 1]\n",
+ "[21, 30, 25, 14, 5, 1]\n",
+ "[51, 76, 69, 44, 20, 6, 1]\n",
+ "[127, 196, 189, 133, 70, 27, 7, 1]\n",
+ "\n",
+ "Flat triangle: [1, 1, 1, 2, 2, 1, 4, 5]\n",
+ "\n",
+ "Inverse triangle:\n",
+ "\n",
+ "[1]\n",
+ "[-1, 1]\n",
+ "[0, -2, 1]\n",
+ "[1, 1, -3, 1]\n",
+ "[-1, 2, 3, -4, 1]\n",
+ "[0, -4, 2, 6, -5, 1]\n",
+ "[1, 2, -9, 0, 10, -6, 1]\n",
+ "[-1, 3, 9, -15, -5, 15, -7, 1]\n",
+ "\n",
+ "Inverse: [1, -1, 1, 0, -2, 1, 1, 1]\n",
+ "\n",
+ "Diagonal triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1]\n",
+ "[2, 1]\n",
+ "[4, 2]\n",
+ "[9, 5, 1]\n",
+ "[21, 12, 3]\n",
+ "[51, 30, 9, 1]\n",
+ "[127, 76, 25, 4]\n",
+ "\n",
+ "Diagonal: [1, 1, 2, 1, 4, 2, 9, 5]\n",
+ "\n",
+ "Polynomial values, array:\n",
+ "\n",
+ "[1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "[1, 2, 3, 4, 5, 6, 7, 8]\n",
+ "[2, 5, 10, 17, 26, 37, 50, 65]\n",
+ "[4, 13, 34, 73, 136, 229, 358, 529]\n",
+ "[9, 35, 117, 315, 713, 1419, 2565, 4307]\n",
+ "[21, 96, 405, 1362, 3741, 8796, 18381, 35070]\n",
+ "[51, 267, 1407, 5895, 19635, 54531, 131727, 285567]\n",
+ "[127, 750, 4899, 25528, 103071, 338082, 944035, 2325324]\n",
+ "\n",
+ "Polynomial values, triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[2, 2, 1]\n",
+ "[4, 5, 3, 1]\n",
+ "[9, 13, 10, 4, 1]\n",
+ "[21, 35, 34, 17, 5, 1]\n",
+ "[51, 96, 117, 73, 26, 6, 1]\n",
+ "[127, 267, 405, 315, 136, 37, 7, 1]\n",
+ "\n",
+ "PolyVal: [1, 1, 1, 2, 2, 1, 4, 5]\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "TraitCard(MotzkinTriangle, dim)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 44,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "=================\n",
+ "JacobsthalTriangle \n",
+ "\n",
+ "Triangle: [1, 1, 1, 1, 2, 1, 3, 5]\n",
+ "Sum: [1, 2, 4, 12, 32, 88, 240, 656]\n",
+ "EvenSum: [1, 1, 2, 6, 16, 44, 120, 328]\n",
+ "OddSum: [0, 1, 2, 6, 16, 44, 120, 328]\n",
+ "AltSum: [1, 0, 0, 0, 0, 0, 0, 0]\n",
+ "DiagSum: [1, 1, 2, 5, 11, 26, 59, 137]\n",
+ "Central: [1, 2, 10, 52]\n",
+ "LeftSide: [1, 1, 1, 3, 5, 11, 21, 43]\n",
+ "RightSide: [1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "PosHalf: [1, 3, 9, 51, 225, 1083, 5049, 23811]\n",
+ "NegHalf: [1, -1, 1, -9, 17, -89, 225, -937]\n",
+ "PolyVal2: [255, 1793, 13053, 100623, 784041, 6167407, 48874041, 389613599]\n",
+ "PolyVal3: [3280, 24604, 186988, 1483608, 11829476, 94793360, 762691476, 6157157152]\n",
+ "N0TS: [0, 1, 4, 14, 48, 156, 496, 1544]\n",
+ "NATS: [1, 3, 8, 26, 80, 244, 736, 2200]\n",
+ "\n",
+ "\n",
+ "Triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[1, 2, 1]\n",
+ "[3, 5, 3, 1]\n",
+ "[5, 12, 10, 4, 1]\n",
+ "[11, 27, 28, 16, 5, 1]\n",
+ "[21, 62, 75, 52, 23, 6, 1]\n",
+ "[43, 137, 193, 159, 85, 31, 7, 1]\n",
+ "\n",
+ "Flat triangle: [1, 1, 1, 1, 2, 1, 3, 5]\n",
+ "\n",
+ "Inverse triangle:\n",
+ "\n",
+ "[1]\n",
+ "[-1, 1]\n",
+ "[1, -2, 1]\n",
+ "[-1, 1, -3, 1]\n",
+ "[1, 4, 2, -4, 1]\n",
+ "[-1, -7, 10, 4, -5, 1]\n",
+ "[1, -14, -25, 16, 7, -6, 1]\n",
+ "[-1, 65, -21, -55, 21, 11, -7, 1]\n",
+ "\n",
+ "Inverse: [1, -1, 1, 1, -2, 1, -1, 1]\n",
+ "\n",
+ "Diagonal triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[3, 2]\n",
+ "[5, 5, 1]\n",
+ "[11, 12, 3]\n",
+ "[21, 27, 10, 1]\n",
+ "[43, 62, 28, 4]\n",
+ "\n",
+ "Diagonal: [1, 1, 1, 1, 3, 2, 5, 5]\n",
+ "\n",
+ "Polynomial values, array:\n",
+ "\n",
+ "[1, 1, 1, 1, 1, 1, 1, 1]\n",
+ "[1, 2, 3, 4, 5, 6, 7, 8]\n",
+ "[1, 4, 9, 16, 25, 36, 49, 64]\n",
+ "[3, 12, 33, 72, 135, 228, 357, 528]\n",
+ "[5, 32, 117, 320, 725, 1440, 2597, 4352]\n",
+ "[11, 88, 417, 1424, 3895, 9096, 18893, 35872]\n",
+ "[21, 240, 1485, 6336, 20925, 57456, 137445, 295680]\n",
+ "[43, 656, 5289, 28192, 112415, 362928, 999901, 2437184]\n",
+ "\n",
+ "Polynomial values, triangle:\n",
+ "\n",
+ "[1]\n",
+ "[1, 1]\n",
+ "[1, 2, 1]\n",
+ "[3, 4, 3, 1]\n",
+ "[5, 12, 9, 4, 1]\n",
+ "[11, 32, 33, 16, 5, 1]\n",
+ "[21, 88, 117, 72, 25, 6, 1]\n",
+ "[43, 240, 417, 320, 135, 36, 7, 1]\n",
+ "\n",
+ "PolyVal: [1, 1, 1, 1, 2, 1, 3, 4]\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "TraitCard(JacobsthalTriangle, dim)"
+ ]
+ }
+ ],
+ "metadata": {
+ "@webio": {
+ "lastCommId": null,
+ "lastKernelId": null
+ },
+ "kernelspec": {
+ "display_name": "Julia 1.6.0",
+ "language": "julia",
+ "name": "julia-1.6"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.6.0"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/docs/make.jl b/docs/make.jl
index eb02a57..dda3519 100644
--- a/docs/make.jl
+++ b/docs/make.jl
@@ -1,4 +1,9 @@
-using Documenter, IntegerTriangles
+prodir = realpath(joinpath(dirname(dirname(@__FILE__))))
+srcdir = joinpath(prodir, "src")
+srcdir ∉ LOAD_PATH && push!(LOAD_PATH, srcdir)
+
+using Documenter
+#using IntegerTriangles
makedocs(
modules = [IntegerTriangles],
@@ -6,6 +11,7 @@ makedocs(
clean = true,
doctest = false,
pages = [
+ "Introduction" => "introduction.md",
"Library" => "index.md",
"Modules" => "modules.md",
]
diff --git a/docs/src/TrianglesLogo.png b/docs/src/TrianglesLogo.png
index 6b50508..8c10bb3 100644
Binary files a/docs/src/TrianglesLogo.png and b/docs/src/TrianglesLogo.png differ
diff --git a/docs/src/index.md b/docs/src/index.md
index 027e78c..97d7f69 100644
--- a/docs/src/index.md
+++ b/docs/src/index.md
@@ -10,6 +10,9 @@ AbstractTriangle
AltSum
```
```@docs
+AssociatedTriangles
+```
+```@docs
BernoulliPolynomial
```
```@docs
@@ -61,6 +64,9 @@ EulerianTransform
EulerianTriangle
```
```@docs
+EulerianTriangle2
+```
+```@docs
Evaluate
```
```@docs
@@ -73,6 +79,9 @@ Explore
ExtCatalanTriangle
```
```@docs
+Factorial
+```
+```@docs
FibonacciTransform
```
```@docs
@@ -85,6 +94,12 @@ Flat
GetSeqnum
```
```@docs
+InvRev
+```
+```@docs
+Inverse
+```
+```@docs
InverseTriangle
```
```@docs
@@ -193,7 +208,13 @@ QTri
RecTriangle
```
```@docs
-ReversedPolynomial
+RevInv
+```
+```@docs
+Reverse
+```
+```@docs
+ReversePolynomial
```
```@docs
RightSide
@@ -232,7 +253,10 @@ StirlingSetTransform
StirlingSetTriangle
```
```@docs
-Sum
+TRAITS
+```
+```@docs
+TRIANGLES
```
```@docs
Telescope
@@ -256,6 +280,9 @@ TransNat1
TransSqrs
```
```@docs
+TransTraits
+```
+```@docs
TransUnos
```
```@docs
@@ -274,6 +301,9 @@ ZSeq
ZTri
```
```@docs
+oeis_notinstalled
+```
+```@docs
oeis_search
```
```@docs
@@ -283,6 +313,9 @@ profilepath
search_failed
```
```@docs
+xTraitCard
+```
+```@docs
ℚInt
```
```@docs
diff --git a/docs/src/introduction.md b/docs/src/introduction.md
new file mode 100644
index 0000000..8b6c8de
--- /dev/null
+++ b/docs/src/introduction.md
@@ -0,0 +1,43 @@
+
+
+# Julia implementations of integer triangles.
+
+We give a framework for computing mathematical integer triangles and use
+it to create so called "Trait Cards".
+
+A trait card is a compilation of the essential characteristics of an integer triangle,
+whereby we understand the characteristics of a triangle to be integer sequences that
+can be obtained from the triangle by elementary transformations.
+
+Overview tables can be automatically generated for a variety of triangles and traits.
+
+| A-Number | Triangle | Form | Function | Sequence |
+|:--------:|:---------- |:----:|:--------- |:------------------------------------------- |
+| A000302 | Binomial | Std | PolyVal3 | 1, 4, 16, 64, 256, 1024, 4096, 16384 |
+| A001333 | SchroederB | Inv | AltSum | 1, -1, 3, -7, 17, -41, 99, -239 |
+| A006012 | SchroederL | Inv | AltSum | 1, -2, 6, -20, 68, -232, 792, -2704 |
+| A026302 | Motzkin | Rev | Central | 1, 2, 9, 44, 230, 1242, 6853, 38376 |
+| A103194 | Laguerre | Std | TransNat0 | 0, 1, 6, 39, 292, 2505, 24306, 263431 |
+| A111884 | Lah | Std | TransAlts | 1, -1, -1, -1, 1, 19, 151, 1091 |
+| nothing | Laguerre | Rev | TransNat1 | 1, 3, 15, 97, 753, 6771, 68983, 783945 |
+
+
+Important: Note that we assume all sequences to start at offset = 0. Also note that all
+references to A-numbers are approximativ only, i.e. the first few terms of the sequence
+may differ and the OEIS-'offset' is always disregarded.
+
+To use this feature you have to download the file [stripped.gz](http://oeis.org/stripped.gz)
+from oeis.org, expand it and put it in the 'data' directory.
+
+To see what you can expect start by executing
+
+ using IntegerTriangles
+ TraitCard(BinomialTriangle, 8)
+
+You can also look at the demo [notebook](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/demos/IntegerTriangles.ipynb).
+
+An introduction to the project can be found in:
+
+* [Tutorial part 1](http://luschny.de/julia/triangles/TutorialTrianglesPart1.html)
+* [Tutorial part 2](http://luschny.de/julia/triangles/TutorialTrianglesPart2.html)
+* [Tutorial part 3](http://luschny.de/julia/triangles/TutorialTrianglesPart3.html)
diff --git a/docs/src/modules.md b/docs/src/modules.md
index 9bf4e38..cccb25f 100644
--- a/docs/src/modules.md
+++ b/docs/src/modules.md
@@ -1,14 +1,25 @@
## 🔶 [TrianglesBase](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/src/TrianglesBase.jl)
-Return the number of permutations of n letters, ``n! = ∏(1, n)``,
-the factorial of ``n``. (Nota: The notation is a shortcut. The use of '!' breaks
-Julia naming conventions, therefore use it only internally.)
+Supertype for sequences (or sequence-like types).
## 🔶 [TrianglesExamples](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/src/TrianglesExamples.jl)
Recurrence for A132393, StirlingCycle numbers.
+## 🔶 [TrianglesTraitCard](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/src/TrianglesTraitCard.jl)
+
+Print the standard traits generated by linear transformation.
+```
+julia> TransTraits(BinomialTriangle)
+BinomialTriangle
+TransUnos: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
+TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
+TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]
+```
+
## 🔶 [TrianglesUtils](https://github.com/OpenLibMathSeq/IntegerTriangles.jl/blob/master/src/TrianglesUtils.jl)
Search the OEIS for a sequence. The file is saved in the 'data' directory in json format.
diff --git a/src/BuildTriangles.jl b/src/BuildTriangles.jl
index da4615c..81bf213 100644
--- a/src/BuildTriangles.jl
+++ b/src/BuildTriangles.jl
@@ -458,9 +458,10 @@ function addsig(srcfile, docfile)
end
nn = nextline(srcfile)
- if !startswith(nn, "const Module")
- println(docfile, "\$(SIGNATURES)")
- end
+ # Do you want signatures?
+ #if !startswith(nn, "const Module")
+ # println(docfile, "\$(SIGNATURES)")
+ #end
println(docfile, n)
n = nn
end
diff --git a/src/IntegerTriangles.jl b/src/IntegerTriangles.jl
index 9ba32dd..2ce10a1 100644
--- a/src/IntegerTriangles.jl
+++ b/src/IntegerTriangles.jl
@@ -1,8 +1,8 @@
# This file is part of IntegerTriangles.
# Copyright Peter Luschny. License is MIT.
-# Version of: UTC 2021-04-14 11:04:40
-# 722dd81e-9d00-11eb-2215-f34eb0d0d280
+# Version of: UTC 2021-04-19 09:45:06
+# 28a16bde-a0e3-11eb-0030-a535525f3034
# Do not edit this file, it is generated from the modules and will be overwritten!
# Edit the modules in the src directory and build this file with BuildTriangles.jl!
@@ -18,6 +18,7 @@ export
AbstractSequence,
AbstractTriangle,
AltSum,
+AssociatedTriangles,
BernoulliPolynomial,
Binomial,
BinomialTransform,
@@ -35,14 +36,18 @@ EgfExpansionCoeff,
EgfExpansionPoly,
EulerianTransform,
EulerianTriangle,
+EulerianTriangle2,
Evaluate,
EvenSum,
Explore,
ExtCatalanTriangle,
+Factorial,
FibonacciTransform,
FibonacciTriangle,
Flat,
GetSeqnum,
+InvRev,
+Inverse,
InverseTriangle,
JacobsthalTransform,
JacobsthalTriangle,
@@ -79,7 +84,9 @@ QPolySeq,
QSeq,
QTri,
RecTriangle,
-ReversedPolynomial,
+RevInv,
+Reverse,
+ReversePolynomial,
RightSide,
RiordanSquare,
SchroederBigTransform,
@@ -92,7 +99,8 @@ StirlingCycleTransform,
StirlingCycleTriangle,
StirlingSetTransform,
StirlingSetTriangle,
-Sum,
+TRAITS,
+TRIANGLES,
Telescope,
TraitCard,
Trans,
@@ -100,15 +108,18 @@ TransAlts,
TransNat0,
TransNat1,
TransSqrs,
+TransTraits,
TransUnos,
ZInt,
ZPolyRing,
ZPolySeq,
ZSeq,
ZTri,
+oeis_notinstalled,
oeis_search,
profilepath,
search_failed,
+xTraitCard,
ℚInt,
ℚPolyRing,
ℚPolySeq,
@@ -127,38 +138,106 @@ I225478,T225478,
I271703,T271703
# *** TrianglesBase.jl ****************
import Base.sum
+"""
+Supertype for sequences (or sequence-like types).
+"""
abstract type AbstractSequence end
+"""
+Supertype for triangles (or triangles-like types).
+"""
abstract type AbstractTriangle end
const Seq{T} = Array{T,1}
-const ℤInt = Nemo.fmpz # (alias for Nemo.fmpz)
-const ℚInt = Nemo.fmpq # (alias for Nemo.fmpq)
-const ℤSeq = Seq{ℤInt} # (alias for Array{fmpz, 1})
-const ℚSeq = Seq{ℚInt} # (alias for Array{fmpq, 1})
-const ℤTri = Seq{ℤSeq} # (alias for Array{Array{fmpz, 1}, 1})
-const ℚTri = Seq{ℚSeq} # (alias for Array{Array{fmpq, 1}, 1})
-ZSeq(len::Int) = ℤSeq(undef, len) # (constructor for Array{fmpz, 1})
-QSeq(len::Int) = ℚSeq(undef, len) # (constructor for Array{fmpq, 1})
-ZSeq(len::Int, f::Function) = [ZZ(f(n)) for n in 0:len-1] # (constructor for Array{fmpz, 1})
-QSeq(len::Int, f::Function) = [QQ(f(n)) for n in 0:len-1] # (constructor for Array{fmpq, 1})
-function ZTri(dim::Int; reg=false) # (constructor for Array{Array{fmpz, 1}, 1})
+"""
+ℤInt is an alias for the type Nemo.fmpz.
+"""
+const ℤInt = Nemo.fmpz
+"""
+ℚInt is an alias for the type Nemo.fmpq.
+"""
+const ℚInt = Nemo.fmpq
+"""
+ℤSeq is an alias for the type Array{Nemo.fmpz, 1}.
+"""
+const ℤSeq = Seq{ℤInt}
+"""
+ℚSeq is an alias for the type Array{Nemo.fmpq, 1}.
+"""
+const ℚSeq = Seq{ℚInt}
+"""
+ℤTri is an alias for the type Array{Array{Nemo.fmpz, 1}, 1}.
+"""
+const ℤTri = Seq{ℤSeq}
+"""
+ℚTri is an alias for the type Array{Array{Nemo.fmpq, 1}, 1}.
+"""
+const ℚTri = Seq{ℚSeq}
+"""
+Constructor for an ℤSeq of length len. If a second parameter f
+is given the sequence is constructed as [ZZ(f(n)) for n in 0:len-1]
+"""
+ZSeq(len::Int) = ℤSeq(undef, len)
+ZSeq(len::Int, f::Function) = [ZZ(f(n)) for n in 0:len-1]
+"""
+Constructor for an ℚSeq of length len. If a second parameter f
+is given the sequence is constructed as [QQ(f(n)) for n in 0:len-1]
+"""
+QSeq(len::Int) = ℚSeq(undef, len)
+QSeq(len::Int, f::Function) = [QQ(f(n)) for n in 0:len-1]
+"""
+Constructor for ZTri.
+"""
+function ZTri(dim::Int; reg=false)
reg ? ZSeq.(1:dim) : ℤTri(undef, dim)
end
-function QTri(dim::Int; reg=false) # (constructor for Array{Array{fmpq, 1}, 1})
-reg ? QSeq.(1:dim) : ℚTri(undef, dim)
-end
ZTri(dim, f::Function) = f.(0:dim-1)
ZTri(T::AbstractTriangle) = [row for row ∈ T]
-const ℤPoly = Nemo.fmpz_poly # (alias for Nemo.fmpz_poly)
-const ℚPoly = Nemo.fmpq_poly # (alias for Nemo.fmpq_poly)
-const ℤPolySeq = Seq{ℤPoly} # (alias for Array{fmpz_poly, 1})
-const ℚPolySeq = Seq{ℚPoly} # (alias for Array{fmpq_poly, 1})
-const ℤPolyTri = Seq{ℤPolySeq} # (alias for Array{Array{fmpz_poly, 1}, 1})
-const ℚPolyTri = Seq{ℚPolySeq} # (alias for Array{Array{fmpq_poly, 1}, 1})
+"""
+Constructor for QTri.
+"""
+function QTri(dim::Int; reg=false)
+reg ? QSeq.(1:dim) : ℚTri(undef, dim)
+end
+"""
+ℤPoly is an alias for the type Nemo.fmpz_poly.
+"""
+const ℤPoly = Nemo.fmpz_poly
+"""
+ℚPoly is an alias for the type Nemo.fmpq_poly.
+"""
+const ℚPoly = Nemo.fmpq_poly
+"""
+ℤPolySeq is an alias for the type Array{Nemo.fmpz_poly, 1}.
+"""
+const ℤPolySeq = Seq{ℤPoly}
+"""
+ℚPolySeq is an alias for the type Array{Nemo.fmpq_poly, 1}.
+"""
+const ℚPolySeq = Seq{ℚPoly}
+"""
+ℤPolyTri is an alias for the type Array{Array{Nemo.fmpz_poly, 1}, 1}.
+"""
+const ℤPolyTri = Seq{ℤPolySeq}
+"""
+ℚPolyTri is an alias for the type Array{Array{Nemo.fmpq_poly, 1}, 1}.
+"""
+const ℚPolyTri = Seq{ℚPolySeq}
ZPolyRing(x) = PolynomialRing(ZZ, x)
QPolyRing(x) = PolynomialRing(QQ, x)
-Base.sum(T::ℤTri) = [sum(row) for row ∈ T]
-Base.sum(T::ℚTri) = [sum(row) for row ∈ T]
const WARNING_ON_NOTINVERTIBLE = false
+"""
+Return the diagonal triangle T(n - k, k) where k in 0:n/2.
+```
+julia> Println.(DiagonalTriangle(MotzkinTriangle(8)))
+[1]
+[1]
+[2, 1]
+[4, 2]
+[9, 5, 1]
+[21, 12, 3]
+[51, 30, 9, 1]
+[127, 76, 25, 4] # A106489
+```
+"""
function DiagonalTriangle(T::ℤTri)
dim = length(T)
U = ZTri(dim)
@@ -171,77 +250,351 @@ U[n] = R
end
U
end
+"""
+The sum of a ℤTri is the sequence of the sum of the rows.
+"""
+Base.sum(T::ℤTri) = [sum(row) for row ∈ T]
+Base.sum(T::ℚTri) = [sum(row) for row ∈ T]
+"""
+The EvenSum of a ℤSeq is the sum of the even indexed terms, indexing starts with 0.
+```
+julia> EvenSum([0, 1, 2, 3, 4, 5])
+0 + 2 + 4 = 6
+```
+"""
EvenSum(A) = sum(A[1:2:end])
+"""
+The OddSum of a ℤSeq is the sum of the odd indexed terms, indexing starts with 0.
+```
+julia> OddSum([0, 1, 2, 3, 4, 5])
+1 + 3 + 5 = 9
+```
+"""
OddSum(A) = sum(A[2:2:end])
+"""
+The AltSum of a ℤSeq is the alternating sum.
+```
+julia> AltSum([0, 1, 2, 3, 4, 5])
++ 0 - 1 + 2 - 3 + 4 - 5 = 6 - 9 = - 3
+```
+"""
AltSum(A) = EvenSum(A) - OddSum(A)
-Middle(A) = A[div(end + 1, 2)]
-LeftSide(A) = A[1]
+"""
+The Middle of a ℤSeq A is the middle term, A[div(begin + end, 2)].
+```
+julia> Middle([0, 1, 2, 3, 4, 5])
+3
+```
+"""
+Middle(A) = A[div(begin + end, 2)]
+"""
+The LeftSide of a ℤSeq is the first term.
+```
+julia> LeftSide([0, 1, 2, 3, 4, 5])
+0
+```
+"""
+LeftSide(A) = A[begin]
+"""
+The RightSide of a ℤSeq is the last term.
+```
+julia> RightSide([0, 1, 2, 3, 4, 5])
+5
+```
+"""
RightSide(A) = A[end]
+"""
+The EvenSum of a ℤTri is the sequence of the even sums of the rows.
+```
EvenSum(T::ℤTri) = EvenSum.(T)
+```
+"""
+EvenSum(T::ℤTri) = EvenSum.(T)
+"""
+The OddSum of a ℤTri is the sequence of the odd sums of the rows.
+```
OddSum(T::ℤTri) = OddSum.(T)
+```
+"""
+OddSum(T::ℤTri) = OddSum.(T)
+"""
+The AltSum of a ℤTri is the sequence of the alternating sums of the rows.
+```
AltSum(T::ℤTri) = EvenSum(T) - OddSum(T)
+```
+"""
+AltSum(T::ℤTri) = EvenSum(T) - OddSum(T)
+"""
+The DiagSum of a ℤTri is the sum of the diagonal triangle.
+```
+DiagSum(T::ℤTri) = sum(DiagonalTriangle(T))
+```
+"""
DiagSum(T::ℤTri) = sum(DiagonalTriangle(T))
+"""
+The Middle of a ℤTri is the sequence of the middle term in the rows.
+```
+Middle(T::ℤTri) = Middle.(T)
+```
+"""
Middle(T::ℤTri) = Middle.(T)
-Central(T::ℤTri) = Middle.(T[1:2:end])
+"""
+The Central of a ℤTri is the sequence of the middle term
+of the even indexed rows, indexing starts with 0.
+```
+Central(T::ℤTri) = Middle.(T[begin:2:end])
+```
+"""
+Central(T::ℤTri) = Middle.(T[begin:2:end])
+"""
+The LeftSide of a ℤTri is the sequence of the first term in the rows.
+```
LeftSide(T::ℤTri) = LeftSide.(T)
+```
+"""
+LeftSide(T::ℤTri) = LeftSide.(T)
+"""
+The RightSide of a ℤTri is the sequence of the last term in the rows.
+```
RightSide(T::ℤTri) = RightSide.(T)
+```
+"""
+RightSide(T::ℤTri) = RightSide.(T)
+"""
+Return the ℤ-polynomial whose coefficients are the terms of the sequence.
+```
+[1, 2, 3] -> 1 + 2*x + 3*x^2
+```
+"""
function Polynomial(S::ℤSeq)
R, x = ZPolyRing("x")
sum(c * x^(k - 1) for (k, c) ∈ enumerate(S))
end
+"""
+Return the sequence of ℤ-polynomials whose coefficients are the terms of the triangle.
+```
+Polynomial(T::ℤTri) = Polynomial.(T)
+```
+"""
Polynomial(T::ℤTri) = Polynomial.(T)
-ReversedPolynomial(S::ℤSeq) = Polynomial(reverse(S))
-ReversedPolynomial(T::ℤTri) = ReversedPolynomial.(T)
-function PolynomialFunction(s)
+"""
+Return the ℤ-polynomial whose coefficients are the terms of the reversed sequence.
+```
+[1, 2, 3] -> x^2 + 2*x + 3
+```
+"""
+ReversePolynomial(S::ℤSeq) = Polynomial(reverse(S))
+"""
+Return the sequence of ℤ-polynomials whose coefficients are the terms of the reversed triangle.
+```
+ReversePolynomial(T::ℤTri) = ReversePolynomial.(T)
+```
+"""
+ReversePolynomial(T::ℤTri) = ReversePolynomial.(T)
+"""
+Return the polynomial function associated with the polynomial with coefficients
+given by the sequence S. A polynomial function evaluates to Float64 values.
+```
+p = PolynomialFunction([1, 2, 3])
+julia> [p(r) for r in 0:3]
+4-element Vector{Float64}:
+ 1.0
+ 6.0
+ 17.0
+ 34.0
+```
+"""
+function PolynomialFunction(S)
y -> sum(Float64(c) * y^(k - 1)
-for (k, c) ∈ enumerate(s))
+for (k, c) ∈ enumerate(S))
end
+"""
+Return the sequence of polynomial functions associated to the polynomials with coefficients
+given by the triangle T.
+```
PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)
-import Nemo.numerator
-numerator(T::ℚTri) = [numerator.(t) for t ∈ T]
-Coefficients(p) = coeff.(p, 0:degree(p))
-Coefficients(P::AbstractArray) = Coefficients.(P)
+```
+"""
+PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)
+"""
+Return the value of the ℤ-polynomial p evaluated at x.
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+Evaluate(p, 2)
+17
+```
+"""
Evaluate(p::ℤPoly, x) = subst(p, x)
Evaluate(p::ℚPoly, x) = subst(p, x)
+"""
+Return the sequence of values to which the sequence of ℤ-polynomials P evaluate at x.
+```
+julia> R, x = ZPolyRing("x")
+P = [sum(k * x^k for k in 0:n) for n in 0:9]
+Evaluate(P, 3) # A289399
+```
+"""
Evaluate(P::ℤPolySeq, x) = Evaluate.(P, x)
Evaluate(P::ℚPolySeq, x) = Evaluate.(P, x)
-function Transpose!(T::ℤTri)
-for n in 1:length(T), k in 1:n
-@inbounds T[n][k], T[k][n] = T[k][n], T[n][k]
-end
-T
-end
+"""
+Return the list of coefficients of the polynomial p (ordered by ascending powers).
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+Coefficients(p)
+```
+"""
+Coefficients(p) = coeff.(p, 0:degree(p))
+"""
+Return the sequence of list of coefficients of the polynomials P.
+"""
+Coefficients(P::AbstractArray) = Coefficients.(P)
+import Nemo.numerator
+numerator(T::ℚTri) = [numerator.(t) for t ∈ T]
+"""
+Return numerator(2^degree(p) * Evaluate(p, QQ(1, 2))).
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+PosHalf(p)
+11
+```
+"""
PosHalf(p) = numerator(2^degree(p) * Evaluate(p, QQ(1, 2)))
+"""
+Return Numerator((-2)^degree(p) * Evaluate(p, QQ(-1, 2)))
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+NegHalf(p)
+3
+```
+"""
NegHalf(p) = numerator((-2)^degree(p) * Evaluate(p, QQ(-1, 2)))
+"""
+Return the sequence generated by broadcasting PosHalf over the
+rows of the triangle interpreted as a polynomials.
+```
+julia> T = [[ZZ(k) for k in 0:n] for n in 1:5]
+PosHalf(Polynomial(T)) # A000295
+5-element ℤSeq
+ 1
+ 4
+ 11
+ 26
+ 57
+```
+"""
PosHalf(T::ℤTri) = PosHalf.(Polynomial(T))
+"""
+Return the sequence generated by broadcasting NegHalf over the
+rows of the triangle interpreted as a polynomials.
+```
NegHalf(T::ℤTri) = NegHalf.(Polynomial(T))
+```
+"""
+NegHalf(T::ℤTri) = NegHalf.(Polynomial(T))
+"""
+Return the sequence generated by broadcasting PosHalf over the
+sequence of polynomials.
+"""
PosHalf(P::ℤPolySeq) = PosHalf.(P)
+"""
+Return the sequence generated by broadcasting NegHalf over the
+sequence of polynomials.
+"""
NegHalf(P::ℤPolySeq) = NegHalf.(P)
-LinMap(F::Function, V::ℤSeq, n) = sum(F(n - 1)[k] * V[k] for k = 1:n)
-LinMap(F::Function, V::ℤSeq) = LinMap(F, V, length(V))
LinMap(M::ℤTri, V::ℤSeq, n) = sum(M[n][k] * V[k] for k = 1:n)
+"""
+LinMap(M::ℤTri, V::ℤSeq) returns the 'matrix times vector' product
+of M and V. Expands up to length(V) and we assume length(M) >= length(V).
+"""
LinMap(M::ℤTri, V::ℤSeq) = (n -> LinMap(M, V, n)).(1:length(V))
+LinMap(F::Function, V::ℤSeq, n) = sum(F(n - 1)[k] * V[k] for k = 1:n)
+"""
+LinMap(F::Function, V::ℤSeq) returns the 'matrix times vector' product
+of a matrix generated by F and V.
+```
+julia> L(n) = [ZZ(k) for k in 0:n]
+[LinMap(L, L(n)) for n in 0:9]
+0
+1
+5
+14
+30
+... # A000330
+```
+"""
+LinMap(F::Function, V::ℤSeq) = LinMap(F, V, length(V))
+"""
+Transform a ℤSeq V by the matrix/vector product by applying LinMap to (M, V).
+Expands up to min(length(M), length(V)).
+"""
Trans(M::ℤTri, V::ℤSeq) = (n -> LinMap(M, V, n)).(1:min(length(M), length(V)))
-TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+"""
+TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+"""
TransAlts(T) = Trans(T, [(-1)^n * ZZ(1) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransAlts(T) = Trans(T, [(-1)^n * ZZ(1) for n = 0:length(T)])
+"""
+TransSqrs(T) = Trans(T, [ZZ(n^2) for n = 0:length(T)])
+
+Standard trait of T.
+"""
TransSqrs(T) = Trans(T, [ZZ(n^2) for n = 0:length(T)])
+"""
TransNat0(T) = Trans(T, [ZZ(n) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransNat0(T) = Trans(T, [ZZ(n) for n = 0:length(T)])
+"""
TransNat1(T) = Trans(T, [ZZ(n) for n = 1:length(T)])
+
+Standard trait of T.
+"""
+TransNat1(T) = Trans(T, [ZZ(n) for n = 1:length(T)])
+"""
+Return an iterator expanding the given sequence to a regular triangle.
+```
+julia> T = Telescope(ℤInt[1, 2, 3, 4, 5, 6])
+collect(T)
+ [1]
+ [1, 2]
+ [1, 2, 3]
+ [1, 2, 3, 4]
+ [1, 2, 3, 4, 5]
+ [1, 2, 3, 4, 5, 6]
+```
+"""
Telescope(A::ℤSeq) = (A[1:k] for k = 1:size(A, 1))
+"""
+Return an iterator expanding the sequence generated by the function f to a regular triangle.
+"""
Telescope(len::Int, f::Function) = Telescope([ZZ(f(k)) for k = 0:len - 1])
"""
-Return the number of permutations of n letters, ``n! = ∏(1, n)``,
-the factorial of ``n``. (Nota: The notation is a shortcut. The use of '!' breaks
-Julia naming conventions, therefore use it only internally.)
-$(SIGNATURES)
+Return the factorial ``n! = ∏(1, n)``.
"""
-F!(n) = Nemo.factorial(ZZ(n))
+Factorial(n) = Nemo.factorial(ZZ(n))
Binomial(n, k) = Nemo.binomial(ZZ(n), ZZ(k))
Binomial(n) = [Binomial(n, k) for k = 0:n]
Binomial(A::ℤSeq) = LinMap(Binomial, A)
BinomialTriangle(dim) = [Binomial(n) for n = 0:dim - 1]
BinomialTransform(A::ℤSeq) = Binomial.(Telescope(A))
Laplace(s, k) = factorial(k) * coeff(s, k)
+"""
+Return the Laplace expansion of a bivariate exponential
+generating function as a power series, or, if 'coeff=true',
+as the coefficients of the series.
+"""
function EgfExpansion(prec, gf::Function, coeff=true)
R, x = QPolyRing("x")
S, t = PowerSeriesRing(R, prec + 1, "t")
@@ -249,7 +602,15 @@ ser = gf(x, t)
P = Laplace.(ser, 0:prec - 1)
coeff ? Coefficients.(P) : P
end
+"""
+Return the Laplace expansion of a bivariate exponential
+generating function as a power series.
+"""
EgfExpansionCoeff(prec, gf::Function) = EgfExpansion(prec, gf, true)
+"""
+Return the Laplace expansion of a bivariate exponential generating
+function as a list of the coefficients of the power series.
+"""
EgfExpansionPoly(prec, gf::Function) = EgfExpansion(prec, gf, false)
function OrthoPoly(dim::Int, s::Function, t::Function)
T = ZTri(dim, reg=true) # regular triangle
@@ -323,7 +684,7 @@ catch e
if isa(e, Exception)
if WARNING_ON_NOTINVERTIBLE
@warn("\n\n Not invertible!\n\n")
-end
+end
return []
end
end
@@ -335,6 +696,19 @@ return []
end
return [[numerator(invM[n, k]) for k = 1:n] for n = 1:dim]
end
+"""
+Alias for InverseTriangle
+"""
+Inverse(T::ℤTri) = InverseTriangle(T)
+"""
+Return the row reversed triangle.
+"""
+Reverse(T::ℤTri) = reverse.(T)
+function RevInv(T::ℤTri)
+I = Inverse(T)
+return I != [] ? Reverse(I) : []
+end
+InvRev(T::ℤTri) = Inverse(Reverse(T))
function Flat(T::ℤTri)
Empty(s) = isempty(s) ? [ZZ(0)] : s
[z for t ∈ T for z ∈ Empty(t)]
@@ -356,23 +730,22 @@ Print(T::ℤSeq) = Println(IOContext(stdout), T, false)
"""
A recursive triangle `RecTriangle` is a subtype of `AbstractTriangle`. The rows
of the triangle are generated by a function `gen(n, k, prevrow)` defined for
-``n ≥ 0`` and ``0 ≤ k ≤ n``. The function returns value of type fmpz.
+``n ≥ 0`` and ``0 ≤ k ≤ n``. The function returns value of type ℤInt.
The parameter prevrow is a function which returns the values of `row(n-1)` of
the triangle and 0 if ``k < 0`` or ``k > n``. The function prevrow is provided
by an instance of `RecTriangle` and must not be defined by the user.
-$(SIGNATURES)
"""
struct RecTriangle <: AbstractTriangle
dim::Int
-A::ℤSeq
+A::ℤSeq
gen::Function # generating function
function RecTriangle(dim::Int, gen::Function)
dim <= 0 && throw(ArgumentError("dim has to be a positive integer."))
new(
dim,
fill(ZZ(0), dim),
-(n::Int, k::Int, f::Function) -> gen(n, k, f)::fmpz,
+(n::Int, k::Int, f::Function) -> gen(n, k, f)::ℤInt,
)
end
end
@@ -383,7 +756,6 @@ T.A[1] = ZZ(top)
end
"""
Return the next row of the triangle.
-$(SIGNATURES)
"""
function Base.iterate(T::RecTriangle, n)
@inline prevrow(A, n) = (k) -> (k < 0 || k > n) ? ZZ(0) : A[k + 1]
@@ -396,7 +768,7 @@ end
(row, n + 1)
end
Base.length(R::RecTriangle) = R.dim
-Base.eltype(R::RecTriangle) = fmpz
+Base.eltype(R::RecTriangle) = ℤInt
function PolyArray(T::ℤTri)
P = Polynomial(T)
dim = length(T)
@@ -422,6 +794,31 @@ P = PolyArray(T)
end
PolyVal2(T::ℤTri) = PolyValue(T, 2)
PolyVal3(T::ℤTri) = PolyValue(T, 3)
+const TRAITS = Function[
+Flat,
+Reverse,
+Inverse,
+RevInv,
+InvRev,
+sum,
+EvenSum,
+OddSum,
+AltSum,
+DiagSum,
+Middle,
+Central,
+LeftSide,
+RightSide,
+PosHalf,
+NegHalf,
+PolyVal2,
+PolyVal3,
+TransUnos,
+TransAlts,
+TransSqrs,
+TransNat0,
+TransNat1
+]
# *** TrianglesExamples.jl ****************
function PrimeDivisors(n)
n < 2 && return ZInt[]
@@ -439,7 +836,7 @@ row[k] = prevrow[k - 1] + prevrow[k] * (n + k - 2)
end
CacheLah[n] = row
end
-LahNumbers(n, k) = LahNumbers(n + 1)[k + 1]
+LahNumbers(n, k) = LahNumbers(n)[k + 1]
function LahTriangle(size)
length(CacheLah) < size && LahNumbers(size)
[CacheLah[n] for n = 0:size - 1]
@@ -494,7 +891,6 @@ ls(n) = isodd(n) ? 2 : 1
SchröderLTriangle(dim) = DelehamΔ(dim, ls, n -> 0^n)
"""
Recurrence for A132393, StirlingCycle numbers.
-$(SIGNATURES)
"""
function R132393(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
@@ -502,7 +898,6 @@ function R132393(n::Int, k::Int, prevrow::Function)
end
"""
Recurrence for A048993, StirlingSet numbers.
-$(SIGNATURES)
"""
function R048993(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
@@ -510,7 +905,6 @@ k * prevrow(k) + prevrow(k - 1)
end
"""
Recurrence for A271703, Lah numbers.
-$(SIGNATURES)
"""
function R271703(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
@@ -518,7 +912,6 @@ function R271703(n::Int, k::Int, prevrow::Function)
end
"""
Recurrence for A094587, (Rising factorials).
-$(SIGNATURES)
"""
function R094587(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
@@ -527,16 +920,14 @@ end
"""
Recurrence for A008279. Number of permutations of n things k at a time.
(Falling factorials)
-$(SIGNATURES)
"""
function R008279(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
prevrow(k) + k * prevrow(k - 1)
end
"""
-Iterates over the first n rows of `A132393`.
+xIterates over the first n rows of `A132393`.
Triangle of unsigned Stirling numbers of the first kind.
-$(SIGNATURES)
"""
I132393(n) = RecTriangle(n, R132393)
T132393(dim) = ZTri(I132393(dim))
@@ -548,7 +939,6 @@ StirlingCycleTransform(A::ℤSeq) = StirlingCycle.(Telescope(A))
"""
Iterates over the first n rows of `A048993`.
Triangle of Stirling numbers of 2nd kind.
-$(SIGNATURES)
"""
I048993(n) = RecTriangle(n, R048993)
T048993(dim) = ZTri(I048993(dim))
@@ -559,16 +949,16 @@ StirlingSet(A::ℤSeq) = LinMap(StirlingSet, A, length(A))
StirlingSetTransform(A::ℤSeq) = StirlingSet.(Telescope(A))
"""
Iterates over the first n rows of `A094587`.
-$(SIGNATURES)
"""
I094587(n) = RecTriangle(n, R094587)
T094587(dim) = ZTri(I094587(dim))
"""
Iterates over the first n rows of `A008279`.
-$(SIGNATURES)
"""
I008279(n) = RecTriangle(n, R008279)
T008279(dim) = ZTri(I008279(dim))
+FallFactTriangle(dim) = ZTri(I008279(dim))
+PermCoeffsTriangle(dim) = ZTri(I008279(dim))
function R225478(n, k, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
4 * prevrow(k - 1) + (4 * n - 1) * prevrow(k)
@@ -622,6 +1012,23 @@ Eulerian(n) = EulerianTriangle(n + 1)[n + 1]
Eulerian(n, k) = Eulerian(n)[k + 1]
Eulerian(A::ℤSeq) = LinMap(Eulerian, A, length(A))
EulerianTransform(A::ℤSeq) = Eulerian.(Telescope(A))
+const CacheEulerian2 = Dict{Tuple{Int,Int},fmpz}()
+function EulerianNumbers2(n, k)
+haskey(CacheEulerian2, (n, k)) && return CacheEulerian2[(n, k)]
+CacheEulerian2[(n, k)] = if (k == n)
+ZZ(1)
+elseif (k <= 0) || (k > n)
+ZZ(0)
+else
+(n - k + 1) * EulerianNumbers2(n - 1, k - 1) +
+(k) * EulerianNumbers2(n - 1, k)
+end
+end
+EulerianTriangle2(dim) = [[EulerianNumbers2(n, k) for k = 0:n] for n = 0:dim - 1]
+Eulerian2(n) = EulerianTriangle2(n + 1)[n + 1]
+Eulerian2(n, k) = Eulerian2(n)[k + 1]
+Eulerian2(A::ℤSeq) = LinMap(Eulerian2, A, length(A))
+EulerianTransform2(A::ℤSeq) = Eulerian2.(Telescope(A))
const CacheNarayana = Dict{Tuple{Int,Int},fmpz}()
function NarayanaNumbers(n::Int, k::Int)
haskey(CacheNarayana, (n, k)) && return CacheNarayana[(n, k)]
@@ -693,37 +1100,56 @@ println("NATS ")
Println(Nut)
trans(Nut) |> Println
end
+const TRIANGLES = Function[
+BinomialTriangle,
+CatalanTriangle,
+EulerianTriangle,
+FibonacciTriangle,
+LaguerreTriangle,
+LahTriangle,
+MotzkinTriangle,
+NarayanaTriangle,
+SchröderBTriangle,
+SchröderLTriangle,
+StirlingCycleTriangle,
+StirlingSetTriangle,
+T008279
+]
# *** TrianglesExplorer.jl ****************
const WARNING_ON_NOTFOUND = false
const Kind = ["Std", "Rev", "Inv", "RevInv", "InvRev"]
const Triangles = LittleDict{String, Function}(
-"Binomial" => BinomialTriangle,
-"Catalan" => CatalanTriangle,
-"Eulerian" => EulerianTriangle,
-"Fibonacci" => FibonacciTriangle,
-"Laguerre" => LaguerreTriangle,
-"Lah" => LahTriangle,
-"Motzkin" => MotzkinTriangle,
-"Narayana" => NarayanaTriangle,
-"SchroederB" => SchröderBTriangle,
+"Binomial" => BinomialTriangle,
+"Catalan" => CatalanTriangle,
+"Eulerian" => EulerianTriangle,
+"Fibonacci" => FibonacciTriangle,
+"Laguerre" => LaguerreTriangle,
+"Lah" => LahTriangle,
+"Motzkin" => MotzkinTriangle,
+"Narayana" => NarayanaTriangle,
+"SchroederB" => SchröderBTriangle,
"SchroederL" => SchröderLTriangle,
-"StirlingCycle" => StirlingCycleTriangle,
-"StirlingSet" => StirlingSetTriangle,
-"PermCoeffs" => T008279
+"StirlingCycle" => StirlingCycleTriangle,
+"StirlingSet" => StirlingSetTriangle,
+"PermCoeffs" => T008279
)
const Traits = LittleDict{String, Function}(
-"Triangle" => Flat,
-"Sum" => sum,
-"EvenSum" => EvenSum,
-"OddSum" => OddSum,
+"Triangle" => Flat,
+"Reverse" => Reverse,
+"Inverse" => Inverse,
+"RevInv" => RevInv,
+"InvRev" => InvRev,
+"Sum" => sum,
+"EvenSum" => EvenSum,
+"OddSum" => OddSum,
"AltSum" => AltSum,
-"DiagSum" => DiagSum,
-"Middle" => Middle,
-"Central" => Central,
-"LeftSide" => LeftSide,
-"RightSide" => RightSide,
-"PosHalf" => PosHalf,
-"NegHalf" => NegHalf,
+"DiagSum" => DiagSum,
+"Middle" => Middle,
+"Central" => Central,
+"LeftSide" => LeftSide,
+"RightSide" => RightSide,
+"PosHalf" => PosHalf,
+"NegHalf" => NegHalf,
"PolyVal2" => PolyVal2,
"PolyVal3" => PolyVal3,
"TransUnos" => TransUnos,
@@ -743,29 +1169,29 @@ print(anum, " ", name, " ", kind, " ", trait, " ")
Println(seq[1:min(8, end)])
end
end
-function TriangleVariant(Tri, dim, kind="Std")
+function TriangleVariant(Tri, dim, kind="Std")
if ! (kind in Kind)
@warn("No valid kind!")
return []
end
M = Tri(dim)
kind == "Std" && return M
-kind == "Rev" && return reverse.(M)
+kind == "Rev" && return reverse.(M)
kind == "InvRev" && (M = reverse.(M))
invM = InverseTriangle(M)
-(kind == "Inv" || kind == "InvRev"
+(kind == "Inv" || kind == "InvRev"
|| invM == []) && return invM
-return reverse.(invM)
+return reverse.(invM)
end
function Explore(triangle, kind, trait, dim)
-T = TriangleVariant(Triangles[triangle], dim, kind)
+T = TriangleVariant(Triangles[triangle], dim, kind)
seq = Traits[trait](T)
Show(stdout, triangle, kind, trait, seq)
end
const LEN = 32
function Explore(triangle, kind, trait)
dim = 32
-T = TriangleVariant(Triangles[triangle], dim, kind)
+T = TriangleVariant(Triangles[triangle], dim, kind)
seq = Traits[trait](T)
anum = GetSeqnum(seq)
anum === nothing && (anum = "nothing")
@@ -776,22 +1202,22 @@ function Explore(trait, dim)
for (name, triangle) in Triangles
for kind in Kind
T = TriangleVariant(triangle, dim, kind)
-if T != []
+if T != []
seq = Traits[trait](T)
Show(stdout, name, kind, trait, seq)
-end
end
end
end
-function Explore(savetofile=false)
+end
+function Explore(savetofile::Bool)
@warn "This will take several minutes and produce the file 'profile.txt' in the data directory."
open(profilepath(), "a") do io
for (name, triangle) in Triangles
for kind in Kind
-T = TriangleVariant(triangle, LEN, kind)
+T = TriangleVariant(triangle, LEN, kind)
if T != []
for (trait, f) in Traits
-Show(io, name, kind, trait, f(T), savetofile)
+Show(io, name, kind, trait, f(T), savetofile)
end
end
end
@@ -816,35 +1242,127 @@ display(P)
end
# *** TrianglesTables.jl ****************
# *** TrianglesTraitCard.jl ****************
+const SEARCH = true
function TraitCard(T::ℚTri, name)
println("WRONG TYPE! Will not process!")
println(name)
println()
end
-function TraitCard(T::ℤTri, name, N)
+function TraitCard(T::ℤTri, name, N, an=false)
+an = an && ! oeis_notinstalled()
println("\n=================")
println(name)
println()
P = Polynomial(T)
-PA = PolyArray(T)
len = max(N, min(N - 1, length(T)))
-S = Flat(T); print("Triangle: "); S[1:len] |> Println
-S = sum(T); print("Sum: "); S[1:len] |> Println
-S = EvenSum(T); print("EvenSum: "); S[1:len] |> Println
-S = OddSum(T); print("OddSum: "); S[1:len] |> Println
-S = AltSum(T); print("AltSum: "); S[1:len] |> Println
-S = DiagSum(T); print("DiagSum: "); S[1:len] |> Println
-S = Central(T); print("Central: "); S[1:min(len, length(S))] |> Println
-S = LeftSide(T); print("LeftSide: "); S[1:len] |> Println
-S = RightSide(T); print("RightSide: "); S[1:len] |> Println
-S = PosHalf(P); print("PosHalf: "); S[1:len] |> Println
-S = NegHalf(P); print("NegHalf: "); S[1:len] |> Println
-S = PolyValue(PA, 2); print("PolyVal2: "); S[1:len] |> Println
-S = PolyValue(PA, 3); print("PolyVal3: "); S[1:len] |> Println
-S = Trans(T, [ZZ(1) for n = 0:32]); print("TransUnos: "); S[1:len] |> Println
-S = Trans(T, [ZZ((-1)^n) for n = 0:32]); print("TransAlts: "); S[1:len] |> Println
-S = Trans(T, [ZZ(n) for n = 0:32]); print("TransNat0: "); S[1:len] |> Println
-S = Trans(T, [ZZ(n) for n = 1:32]); print("TransNat1: "); S[1:len] |> Println
+len = len < 10 ? len : 10
+S = Flat(T); print("Triangle: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+R = Reverse(T)
+FR = Flat(R); print("Reverse: ")
+an && print(GetSeqnum(FR, SEARCH), " ")
+FR[1:len] |> Println
+I = Inverse(T)
+if I != []
+FI = Flat(I); print("Inverse: ")
+an && print(GetSeqnum(FI, SEARCH), " ")
+FI[1:len] |> Println
+RI = Reverse(I)
+RI = Flat(RI); print("RevInv: ")
+an && print(GetSeqnum(RI, SEARCH), " ")
+RI[1:len] |> Println
+end
+IR = Inverse(R)
+if IR != []
+IR = Flat(IR); print("InvRev: ")
+an && print(GetSeqnum(IR, SEARCH), " ")
+IR[1:len] |> Println
+end
+S = DiagonalTriangle(T)
+S = Flat(S);
+print("Diagonal: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = sum(T); print("Sum: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = EvenSum(T); print("EvenSum: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = OddSum(T); print("OddSum: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = AltSum(T); print("AltSum: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = DiagSum(T); print("DiagSum: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Central(T); print("Central: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:min(len, length(S))] |> Println
+S = LeftSide(T); print("LeftSide: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = RightSide(T); print("RightSide: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = PosHalf(P); print("PosHalf: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = NegHalf(P); print("NegHalf: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Trans(T, [ZZ(1) for n = 0:32]); print("TransUnos: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Trans(T, [ZZ((-1)^n) for n = 0:32]); print("TransAlts: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Trans(T, [ZZ(n^2) for n = 0:32]); print("TransSqrs: ");
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Trans(T, [ZZ(n) for n = 0:32]); print("TransNat0: ");
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = Trans(T, [ZZ(n) for n = 1:32]); print("TransNat1: ");
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = PolyVal2(T); print("PolyVal2: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+S = PolyVal3(T); print("PolyVal3: ")
+an && print(GetSeqnum(S, SEARCH), " ")
+S[1:len] |> Println
+end
+"""
+Print the standard traits generated by linear transformation.
+```
+julia> TransTraits(BinomialTriangle)
+BinomialTriangle
+TransUnos: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
+TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
+TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]
+```
+"""
+function TransTraits(triangle::Function)
+dim = 32
+T = triangle(dim)
+println("$triangle ")
+len = min(10, min(dim - 1, length(T)))
+S = Trans(T, [ZZ(1) for n = 0:dim]); anum = GetSeqnum(S, false)
+print("TransUnos: ", anum, " "); S[1:len] |> Println
+S = Trans(T, [ZZ((-1)^n) for n = 0:dim]); anum = GetSeqnum(S, false)
+print("TransAlts: ", anum, " "); S[1:len] |> Println
+S = Trans(T, [ZZ(n^2) for n = 0:dim]); anum = GetSeqnum(S, false)
+print("TransSqrs: ", anum, " "); S[1:len] |> Println
+S = Trans(T, [ZZ(n) for n = 0:dim]); anum = GetSeqnum(S, false)
+print("TransNat0: ", anum, " "); S[1:len] |> Println
+S = Trans(T, [ZZ(n) for n = 1:dim]); anum = GetSeqnum(S, false)
+print("TransNat1: ", anum, " "); S[1:len] |> Println
println()
end
function AssociatedTriangles(T::ℤTri, N)
@@ -872,13 +1390,50 @@ S = PolyTriangle(T)
Println.(S[1:len])
println()
S = Flat(S); print("PolyVal: "); S[1:len] |> Println
-println()
end
-function TraitCard(triangle::Function, dim=8)
+"""
+Prints a list of traits of the triangle without Anums
+and, if assoc=true, also a list of associated triangles.
+"""
+function TraitCard(triangle::Function, dim=8, assoc=false)
T = triangle(dim)
TraitCard(T, "$triangle ", dim)
AssociatedTriangles(T, dim)
end
+"""
+Prints a list of traits of the triangle with Anums.
+xTraitCard is slower and uses much more resources than TraitCard.
+```
+julia> xTraitCard(LaguerreTriangle)
+Triangle: A021009 [1, 1, 1, 2, 4, 1, 6, 18, 9, 1]
+Reverse: A021010 [1, 1, 1, 1, 4, 2, 1, 9, 18, 6]
+Inverse: A021009 [1, -1, 1, 2, -4, 1, -6, 18, -9, 1]
+RevInv: A021010 [1, 1, -1, 1, -4, 2, 1, -9, 18, -6]
+Diagonal: A084950 [1, 1, 2, 1, 6, 4, 24, 18, 1, 120]
+Sum: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+EvenSum: A331325 [1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257]
+OddSum: A331326 [0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857]
+AltSum: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+DiagSum: C001040 [1, 1, 3, 10, 43, 225, 1393, 9976, 81201, 740785]
+Central: A295383 [1, 4, 72, 2400, 117600, 7620480, 614718720, 59364264960]
+LeftSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025167 [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089]
+NegHalf: A025166 [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063, 84149983]
+TransUnos: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+TransAlts: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+TransSqrs: A105219 [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049]
+TransNat0: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat1: C000262 [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091]
+PolyVal2: A087912 [1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856]
+PolyVal3: A277382 [1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449]
+```
+"""
+function xTraitCard(triangle::Function)
+dim = 32
+T = triangle(dim)
+TraitCard(T, "$triangle ", dim, true)
+end
# *** TrianglesUtils.jl ****************
const srcdir = realpath(joinpath(dirname(@__FILE__)))
const ROOTDIR = dirname(srcdir)
@@ -912,7 +1467,6 @@ false
end
"""
Search the OEIS for a sequence. The file is saved in the 'data' directory in json format.
-$(SIGNATURES)
"""
function oeis_search(seq)
seqstr = SeqToString(seq[1:min(end,12)])
@@ -953,12 +1507,10 @@ str *= string(abs(term)) * separator
end
str
end
-const minlen = 30 # fragil!
+const minlen = 30 # fragil! do not reduce!
function GetSeqnum(seq::ℤSeq)
str = SeqToString(seq)
soff = 1; loff = 10
-println(seq)
-println(str)
for ln ∈ eachline(oeis_file())
ln[1] == '#' && continue
l = replace(ln, "-" => "")
@@ -1017,7 +1569,6 @@ end
"""
Print the array ``A`` in the format ``n ↦ A[n]`` for n in the given range.
-$(SIGNATURES)
"""
function MappedShow(A::Array, R::AbstractRange, offset=0)
for k ∈ R
@@ -1031,7 +1582,6 @@ end
"""
Print an integer triangle without typeinfo.
-$(SIGNATURES)
"""
function Show(T::ℤTri, format="std")
if format == "std" # default
diff --git a/src/TrianglesBase.jl b/src/TrianglesBase.jl
index aef5ffd..c003145 100644
--- a/src/TrianglesBase.jl
+++ b/src/TrianglesBase.jl
@@ -9,80 +9,156 @@ using Nemo
import Base.sum
export AbstractSequence, AbstractTriangle
-export ℤInt, ℤSeq, ℤTri, ℚInt, ℚSeq, ℚTri, ℤPolySeq, ℚPolySeq
-export ℤPolyRing, ℚPolyRing
-export ZInt, ZSeq, ZTri, QInt, QSeq, QTri, ZPolySeq, QPolySeq
-export ZPolyRing, QPolyRing
+export ℤInt, ℤSeq, ℤTri, ℚInt, ℚSeq, ℚTri
+export ℤPolySeq, ℚPolySeq, ℤPolyRing, ℚPolyRing
+export ZInt, ZSeq, ZTri, QInt, QSeq, QTri
+export ZPolySeq, QPolySeq, ZPolyRing, QPolyRing
export Polynomial, Evaluate, PolyArray, PolyTriangle
export DiagonalTriangle, OrthoPoly, DelehamΔ
-export EgfExpansionCoeff, EgfExpansionPoly
-export RecTriangle, InverseTriangle
-export PolynomialFunction, ReversedPolynomial, PolyValue, PolyVal2, PolyVal3
+export EgfExpansionCoeff, EgfExpansionPoly, RecTriangle
+export InverseTriangle, Inverse, Reverse, InvRev, RevInv
+export PolynomialFunction, ReversePolynomial, PolyValue
+export PolyVal2, PolyVal3, Coefficients
export RiordanSquare, Println, Print, Telescope, LinMap, Trans
-export Sum, EvenSum, OddSum, AltSum, DiagSum, Central, Middle
+export EvenSum, OddSum, AltSum, DiagSum, Central, Middle
export LeftSide, RightSide, PosHalf, NegHalf, Flat
-export Binomial, BinomialTransform, BinomialTriangle, Coefficients
+export Factorial, Binomial, BinomialTransform, BinomialTriangle
export TransUnos, TransAlts, TransSqrs, TransNat0, TransNat1
+export TRAITS
-# include("TrianglesTypes.jl")
-
+"""
+Supertype for sequences (or sequence-like types).
+"""
abstract type AbstractSequence end
+
+"""
+Supertype for triangles (or triangles-like types).
+"""
abstract type AbstractTriangle end
const Seq{T} = Array{T,1}
-const ℤInt = Nemo.fmpz # (alias for Nemo.fmpz)
-const ℚInt = Nemo.fmpq # (alias for Nemo.fmpq)
+"""
+ℤInt is an alias for the type Nemo.fmpz.
+"""
+const ℤInt = Nemo.fmpz
-const ℤSeq = Seq{ℤInt} # (alias for Array{fmpz, 1})
-const ℚSeq = Seq{ℚInt} # (alias for Array{fmpq, 1})
+"""
+ℚInt is an alias for the type Nemo.fmpq.
+"""
+const ℚInt = Nemo.fmpq
-const ℤTri = Seq{ℤSeq} # (alias for Array{Array{fmpz, 1}, 1})
-const ℚTri = Seq{ℚSeq} # (alias for Array{Array{fmpq, 1}, 1})
+"""
+ℤSeq is an alias for the type Array{Nemo.fmpz, 1}.
+"""
+const ℤSeq = Seq{ℤInt}
+
+"""
+ℚSeq is an alias for the type Array{Nemo.fmpq, 1}.
+"""
+const ℚSeq = Seq{ℚInt}
+
+"""
+ℤTri is an alias for the type Array{Array{Nemo.fmpz, 1}, 1}.
+"""
+const ℤTri = Seq{ℤSeq}
+
+"""
+ℚTri is an alias for the type Array{Array{Nemo.fmpq, 1}, 1}.
+"""
+const ℚTri = Seq{ℚSeq}
# -------------
# ZZ(n) # (constructor for fmpz, defined in Nemo)
# QQ(n, k) # (constructor for fmpq, defined in Nemo)
-ZSeq(len::Int) = ℤSeq(undef, len) # (constructor for Array{fmpz, 1})
-QSeq(len::Int) = ℚSeq(undef, len) # (constructor for Array{fmpq, 1})
+"""
+Constructor for an ℤSeq of length len. If a second parameter f
+is given the sequence is constructed as [ZZ(f(n)) for n in 0:len-1]
+"""
+ZSeq(len::Int) = ℤSeq(undef, len)
+
+ZSeq(len::Int, f::Function) = [ZZ(f(n)) for n in 0:len-1]
-ZSeq(len::Int, f::Function) = [ZZ(f(n)) for n in 0:len-1] # (constructor for Array{fmpz, 1})
-QSeq(len::Int, f::Function) = [QQ(f(n)) for n in 0:len-1] # (constructor for Array{fmpq, 1})
+"""
+Constructor for an ℚSeq of length len. If a second parameter f
+is given the sequence is constructed as [QQ(f(n)) for n in 0:len-1]
+"""
+QSeq(len::Int) = ℚSeq(undef, len)
-function ZTri(dim::Int; reg=false) # (constructor for Array{Array{fmpz, 1}, 1})
+QSeq(len::Int, f::Function) = [QQ(f(n)) for n in 0:len-1]
+
+"""
+Constructor for ZTri.
+"""
+function ZTri(dim::Int; reg=false)
reg ? ZSeq.(1:dim) : ℤTri(undef, dim)
end
-function QTri(dim::Int; reg=false) # (constructor for Array{Array{fmpq, 1}, 1})
+ZTri(dim, f::Function) = f.(0:dim-1)
+ZTri(T::AbstractTriangle) = [row for row ∈ T]
+
+"""
+Constructor for QTri.
+"""
+function QTri(dim::Int; reg=false)
reg ? QSeq.(1:dim) : ℚTri(undef, dim)
end
-ZTri(dim, f::Function) = f.(0:dim-1)
-ZTri(T::AbstractTriangle) = [row for row ∈ T]
-
# ---------------
-const ℤPoly = Nemo.fmpz_poly # (alias for Nemo.fmpz_poly)
-const ℚPoly = Nemo.fmpq_poly # (alias for Nemo.fmpq_poly)
+"""
+ℤPoly is an alias for the type Nemo.fmpz_poly.
+"""
+const ℤPoly = Nemo.fmpz_poly
+
+"""
+ℚPoly is an alias for the type Nemo.fmpq_poly.
+"""
+const ℚPoly = Nemo.fmpq_poly
+
+"""
+ℤPolySeq is an alias for the type Array{Nemo.fmpz_poly, 1}.
+"""
+const ℤPolySeq = Seq{ℤPoly}
+
+"""
+ℚPolySeq is an alias for the type Array{Nemo.fmpq_poly, 1}.
+"""
+const ℚPolySeq = Seq{ℚPoly}
-const ℤPolySeq = Seq{ℤPoly} # (alias for Array{fmpz_poly, 1})
-const ℚPolySeq = Seq{ℚPoly} # (alias for Array{fmpq_poly, 1})
+"""
+ℤPolyTri is an alias for the type Array{Array{Nemo.fmpz_poly, 1}, 1}.
+"""
+const ℤPolyTri = Seq{ℤPolySeq}
-const ℤPolyTri = Seq{ℤPolySeq} # (alias for Array{Array{fmpz_poly, 1}, 1})
-const ℚPolyTri = Seq{ℚPolySeq} # (alias for Array{Array{fmpq_poly, 1}, 1})
+"""
+ℚPolyTri is an alias for the type Array{Array{Nemo.fmpq_poly, 1}, 1}.
+"""
+const ℚPolyTri = Seq{ℚPolySeq}
# ---------------
ZPolyRing(x) = PolynomialRing(ZZ, x)
QPolyRing(x) = PolynomialRing(QQ, x)
-Base.sum(T::ℤTri) = [sum(row) for row ∈ T]
-Base.sum(T::ℚTri) = [sum(row) for row ∈ T]
-
const WARNING_ON_NOTINVERTIBLE = false
+"""
+Return the diagonal triangle T(n - k, k) where k in 0:n/2.
+```
+julia> Println.(DiagonalTriangle(MotzkinTriangle(8)))
+[1]
+[1]
+[2, 1]
+[4, 2]
+[9, 5, 1]
+[21, 12, 3]
+[51, 30, 9, 1]
+[127, 76, 25, 4] # A106489
+```
+"""
function DiagonalTriangle(T::ℤTri)
dim = length(T)
U = ZTri(dim)
@@ -96,90 +172,386 @@ function DiagonalTriangle(T::ℤTri)
U
end
+"""
+The sum of a ℤTri is the sequence of the sum of the rows.
+"""
+Base.sum(T::ℤTri) = [sum(row) for row ∈ T]
+Base.sum(T::ℚTri) = [sum(row) for row ∈ T]
+
+"""
+The EvenSum of a ℤSeq is the sum of the even indexed terms, indexing starts with 0.
+```
+julia> EvenSum([0, 1, 2, 3, 4, 5])
+0 + 2 + 4 = 6
+```
+"""
EvenSum(A) = sum(A[1:2:end])
+
+"""
+The OddSum of a ℤSeq is the sum of the odd indexed terms, indexing starts with 0.
+```
+julia> OddSum([0, 1, 2, 3, 4, 5])
+1 + 3 + 5 = 9
+```
+"""
OddSum(A) = sum(A[2:2:end])
+
+"""
+The AltSum of a ℤSeq is the alternating sum.
+```
+julia> AltSum([0, 1, 2, 3, 4, 5])
++ 0 - 1 + 2 - 3 + 4 - 5 = 6 - 9 = - 3
+```
+"""
AltSum(A) = EvenSum(A) - OddSum(A)
-Middle(A) = A[div(end + 1, 2)]
-LeftSide(A) = A[1]
+
+"""
+The Middle of a ℤSeq A is the middle term, A[div(begin + end, 2)].
+```
+julia> Middle([0, 1, 2, 3, 4, 5])
+3
+```
+"""
+Middle(A) = A[div(begin + end, 2)]
+
+"""
+The LeftSide of a ℤSeq is the first term.
+```
+julia> LeftSide([0, 1, 2, 3, 4, 5])
+0
+```
+"""
+LeftSide(A) = A[begin]
+
+"""
+The RightSide of a ℤSeq is the last term.
+```
+julia> RightSide([0, 1, 2, 3, 4, 5])
+5
+```
+"""
RightSide(A) = A[end]
+"""
+The EvenSum of a ℤTri is the sequence of the even sums of the rows.
+```
EvenSum(T::ℤTri) = EvenSum.(T)
+```
+"""
+EvenSum(T::ℤTri) = EvenSum.(T)
+
+"""
+The OddSum of a ℤTri is the sequence of the odd sums of the rows.
+```
OddSum(T::ℤTri) = OddSum.(T)
+```
+"""
+OddSum(T::ℤTri) = OddSum.(T)
+
+"""
+The AltSum of a ℤTri is the sequence of the alternating sums of the rows.
+```
+AltSum(T::ℤTri) = EvenSum(T) - OddSum(T)
+```
+"""
AltSum(T::ℤTri) = EvenSum(T) - OddSum(T)
+
+"""
+The DiagSum of a ℤTri is the sum of the diagonal triangle.
+```
+DiagSum(T::ℤTri) = sum(DiagonalTriangle(T))
+```
+"""
DiagSum(T::ℤTri) = sum(DiagonalTriangle(T))
+
+"""
+The Middle of a ℤTri is the sequence of the middle term in the rows.
+```
+Middle(T::ℤTri) = Middle.(T)
+```
+"""
Middle(T::ℤTri) = Middle.(T)
-Central(T::ℤTri) = Middle.(T[1:2:end])
+
+"""
+The Central of a ℤTri is the sequence of the middle term
+of the even indexed rows, indexing starts with 0.
+```
+Central(T::ℤTri) = Middle.(T[begin:2:end])
+```
+"""
+Central(T::ℤTri) = Middle.(T[begin:2:end])
+
+"""
+The LeftSide of a ℤTri is the sequence of the first term in the rows.
+```
+LeftSide(T::ℤTri) = LeftSide.(T)
+```
+"""
LeftSide(T::ℤTri) = LeftSide.(T)
+
+"""
+The RightSide of a ℤTri is the sequence of the last term in the rows.
+```
+RightSide(T::ℤTri) = RightSide.(T)
+```
+"""
RightSide(T::ℤTri) = RightSide.(T)
-# Triangles -> Polynomials
+"""
+Return the ℤ-polynomial whose coefficients are the terms of the sequence.
+```
+[1, 2, 3] -> 1 + 2*x + 3*x^2
+```
+"""
function Polynomial(S::ℤSeq)
R, x = ZPolyRing("x")
sum(c * x^(k - 1) for (k, c) ∈ enumerate(S))
end
+
+"""
+Return the sequence of ℤ-polynomials whose coefficients are the terms of the triangle.
+```
+Polynomial(T::ℤTri) = Polynomial.(T)
+```
+"""
Polynomial(T::ℤTri) = Polynomial.(T)
-ReversedPolynomial(S::ℤSeq) = Polynomial(reverse(S))
-ReversedPolynomial(T::ℤTri) = ReversedPolynomial.(T)
+"""
+Return the ℤ-polynomial whose coefficients are the terms of the reversed sequence.
+```
+[1, 2, 3] -> x^2 + 2*x + 3
+```
+"""
+ReversePolynomial(S::ℤSeq) = Polynomial(reverse(S))
+
+"""
+Return the sequence of ℤ-polynomials whose coefficients are the terms of the reversed triangle.
+```
+ReversePolynomial(T::ℤTri) = ReversePolynomial.(T)
+```
+"""
+ReversePolynomial(T::ℤTri) = ReversePolynomial.(T)
-# Triangles -> PolynomialFunctions
-function PolynomialFunction(s)
+"""
+Return the polynomial function associated with the polynomial with coefficients
+given by the sequence S. A polynomial function evaluates to Float64 values.
+```
+p = PolynomialFunction([1, 2, 3])
+julia> [p(r) for r in 0:3]
+4-element Vector{Float64}:
+ 1.0
+ 6.0
+ 17.0
+ 34.0
+```
+"""
+function PolynomialFunction(S)
y -> sum(Float64(c) * y^(k - 1)
- for (k, c) ∈ enumerate(s))
+ for (k, c) ∈ enumerate(S))
end
-PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)
-import Nemo.numerator
-numerator(T::ℚTri) = [numerator.(t) for t ∈ T]
-
-# Polynomials -> Triangles
-Coefficients(p) = coeff.(p, 0:degree(p))
-Coefficients(P::AbstractArray) = Coefficients.(P)
+"""
+Return the sequence of polynomial functions associated to the polynomials with coefficients
+given by the triangle T.
+```
+PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)
+```
+"""
+PolynomialFunction(T::ℤTri) = PolynomialFunction.(T)
+"""
+Return the value of the ℤ-polynomial p evaluated at x.
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+Evaluate(p, 2)
+17
+```
+"""
Evaluate(p::ℤPoly, x) = subst(p, x)
Evaluate(p::ℚPoly, x) = subst(p, x)
+
+"""
+Return the sequence of values to which the sequence of ℤ-polynomials P evaluate at x.
+```
+julia> R, x = ZPolyRing("x")
+P = [sum(k * x^k for k in 0:n) for n in 0:9]
+Evaluate(P, 3) # A289399
+```
+"""
Evaluate(P::ℤPolySeq, x) = Evaluate.(P, x)
Evaluate(P::ℚPolySeq, x) = Evaluate.(P, x)
-function Transpose!(T::ℤTri)
- for n in 1:length(T), k in 1:n
- @inbounds T[n][k], T[k][n] = T[k][n], T[n][k]
- end
- T
-end
+"""
+Return the list of coefficients of the polynomial p (ordered by ascending powers).
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+Coefficients(p)
+```
+"""
+Coefficients(p) = coeff.(p, 0:degree(p))
+"""
+Return the sequence of list of coefficients of the polynomials P.
+"""
+Coefficients(P::AbstractArray) = Coefficients.(P)
+
+import Nemo.numerator
+numerator(T::ℚTri) = [numerator.(t) for t ∈ T]
+
+"""
+Return numerator(2^degree(p) * Evaluate(p, QQ(1, 2))).
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+PosHalf(p)
+11
+```
+"""
PosHalf(p) = numerator(2^degree(p) * Evaluate(p, QQ(1, 2)))
+
+"""
+Return Numerator((-2)^degree(p) * Evaluate(p, QQ(-1, 2)))
+```
+julia> R, x = ZPolyRing("x")
+p = 1 + 2*x + 3*x^2
+NegHalf(p)
+3
+```
+"""
NegHalf(p) = numerator((-2)^degree(p) * Evaluate(p, QQ(-1, 2)))
+"""
+Return the sequence generated by broadcasting PosHalf over the
+rows of the triangle interpreted as a polynomials.
+```
+julia> T = [[ZZ(k) for k in 0:n] for n in 1:5]
+PosHalf(Polynomial(T)) # A000295
+5-element ℤSeq
+ 1
+ 4
+ 11
+ 26
+ 57
+```
+"""
PosHalf(T::ℤTri) = PosHalf.(Polynomial(T))
+
+"""
+Return the sequence generated by broadcasting NegHalf over the
+rows of the triangle interpreted as a polynomials.
+```
+NegHalf(T::ℤTri) = NegHalf.(Polynomial(T))
+```
+"""
NegHalf(T::ℤTri) = NegHalf.(Polynomial(T))
+"""
+Return the sequence generated by broadcasting PosHalf over the
+sequence of polynomials.
+"""
PosHalf(P::ℤPolySeq) = PosHalf.(P)
+
+"""
+Return the sequence generated by broadcasting NegHalf over the
+sequence of polynomials.
+"""
NegHalf(P::ℤPolySeq) = NegHalf.(P)
+LinMap(M::ℤTri, V::ℤSeq, n) = sum(M[n][k] * V[k] for k = 1:n)
+
+"""
+LinMap(M::ℤTri, V::ℤSeq) returns the 'matrix times vector' product
+of M and V. Expands up to length(V) and we assume length(M) >= length(V).
+"""
+LinMap(M::ℤTri, V::ℤSeq) = (n -> LinMap(M, V, n)).(1:length(V))
LinMap(F::Function, V::ℤSeq, n) = sum(F(n - 1)[k] * V[k] for k = 1:n)
+
+"""
+LinMap(F::Function, V::ℤSeq) returns the 'matrix times vector' product
+of a matrix generated by F and V.
+```
+julia> L(n) = [ZZ(k) for k in 0:n]
+[LinMap(L, L(n)) for n in 0:9]
+0
+1
+5
+14
+30
+... # A000330
+```
+"""
LinMap(F::Function, V::ℤSeq) = LinMap(F, V, length(V))
-LinMap(M::ℤTri, V::ℤSeq, n) = sum(M[n][k] * V[k] for k = 1:n)
-LinMap(M::ℤTri, V::ℤSeq) = (n -> LinMap(M, V, n)).(1:length(V))
+"""
+Transform a ℤSeq V by the matrix/vector product by applying LinMap to (M, V).
+Expands up to min(length(M), length(V)).
+"""
Trans(M::ℤTri, V::ℤSeq) = (n -> LinMap(M, V, n)).(1:min(length(M), length(V)))
-TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+"""
+TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransUnos(T) = Trans(T, [ZZ(1) for n = 0:length(T)])
+
+"""
+TransAlts(T) = Trans(T, [(-1)^n * ZZ(1) for n = 0:length(T)])
+
+Standard trait of T.
+"""
TransAlts(T) = Trans(T, [(-1)^n * ZZ(1) for n = 0:length(T)])
+
+"""
TransSqrs(T) = Trans(T, [ZZ(n^2) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransSqrs(T) = Trans(T, [ZZ(n^2) for n = 0:length(T)])
+
+"""
TransNat0(T) = Trans(T, [ZZ(n) for n = 0:length(T)])
+
+Standard trait of T.
+"""
+TransNat0(T) = Trans(T, [ZZ(n) for n = 0:length(T)])
+
+"""
+TransNat1(T) = Trans(T, [ZZ(n) for n = 1:length(T)])
+
+Standard trait of T.
+"""
TransNat1(T) = Trans(T, [ZZ(n) for n = 1:length(T)])
+"""
+Return an iterator expanding the given sequence to a regular triangle.
+```
+julia> T = Telescope(ℤInt[1, 2, 3, 4, 5, 6])
+collect(T)
+ [1]
+ [1, 2]
+ [1, 2, 3]
+ [1, 2, 3, 4]
+ [1, 2, 3, 4, 5]
+ [1, 2, 3, 4, 5, 6]
+```
+"""
Telescope(A::ℤSeq) = (A[1:k] for k = 1:size(A, 1))
+
+"""
+Return an iterator expanding the sequence generated by the function f to a regular triangle.
+"""
Telescope(len::Int, f::Function) = Telescope([ZZ(f(k)) for k = 0:len - 1])
"""
-Return the number of permutations of n letters, ``n! = ∏(1, n)``,
-the factorial of ``n``. (Nota: The notation is a shortcut. The use of '!' breaks
-Julia naming conventions, therefore use it only internally.)
+Return the factorial ``n! = ∏(1, n)``.
"""
-F!(n) = Nemo.factorial(ZZ(n))
+Factorial(n) = Nemo.factorial(ZZ(n))
+
+
Binomial(n, k) = Nemo.binomial(ZZ(n), ZZ(k))
Binomial(n) = [Binomial(n, k) for k = 0:n]
Binomial(A::ℤSeq) = LinMap(Binomial, A)
@@ -189,7 +561,11 @@ BinomialTransform(A::ℤSeq) = Binomial.(Telescope(A))
Laplace(s, k) = factorial(k) * coeff(s, k)
-# Keep module-intern
+"""
+Return the Laplace expansion of a bivariate exponential
+generating function as a power series, or, if 'coeff=true',
+as the coefficients of the series.
+"""
function EgfExpansion(prec, gf::Function, coeff=true)
R, x = QPolyRing("x")
S, t = PowerSeriesRing(R, prec + 1, "t")
@@ -198,7 +574,16 @@ function EgfExpansion(prec, gf::Function, coeff=true)
coeff ? Coefficients.(P) : P
end
+"""
+Return the Laplace expansion of a bivariate exponential
+generating function as a power series.
+"""
EgfExpansionCoeff(prec, gf::Function) = EgfExpansion(prec, gf, true)
+
+"""
+Return the Laplace expansion of a bivariate exponential generating
+function as a list of the coefficients of the power series.
+"""
EgfExpansionPoly(prec, gf::Function) = EgfExpansion(prec, gf, false)
function OrthoPoly(dim::Int, s::Function, t::Function)
@@ -279,7 +664,7 @@ function InverseTriangle(T)
if isa(e, Exception)
if WARNING_ON_NOTINVERTIBLE
@warn("\n\n Not invertible!\n\n")
- end
+ end
return []
end
end
@@ -295,6 +680,24 @@ function InverseTriangle(T)
return [[numerator(invM[n, k]) for k = 1:n] for n = 1:dim]
end
+"""
+Alias for InverseTriangle
+"""
+Inverse(T::ℤTri) = InverseTriangle(T)
+
+"""
+Return the row reversed triangle.
+"""
+Reverse(T::ℤTri) = reverse.(T)
+
+function RevInv(T::ℤTri)
+ I = Inverse(T)
+ return I != [] ? Reverse(I) : []
+end
+
+InvRev(T::ℤTri) = Inverse(Reverse(T))
+
+
function Flat(T::ℤTri)
Empty(s) = isempty(s) ? [ZZ(0)] : s
[z for t ∈ T for z ∈ Empty(t)]
@@ -323,7 +726,7 @@ Print(T::ℤSeq) = Println(IOContext(stdout), T, false)
"""
A recursive triangle `RecTriangle` is a subtype of `AbstractTriangle`. The rows
of the triangle are generated by a function `gen(n, k, prevrow)` defined for
-``n ≥ 0`` and ``0 ≤ k ≤ n``. The function returns value of type fmpz.
+``n ≥ 0`` and ``0 ≤ k ≤ n``. The function returns value of type ℤInt.
The parameter prevrow is a function which returns the values of `row(n-1)` of
the triangle and 0 if ``k < 0`` or ``k > n``. The function prevrow is provided
@@ -331,14 +734,14 @@ by an instance of `RecTriangle` and must not be defined by the user.
"""
struct RecTriangle <: AbstractTriangle
dim::Int
- A::ℤSeq
+ A::ℤSeq
gen::Function # generating function
function RecTriangle(dim::Int, gen::Function)
dim <= 0 && throw(ArgumentError("dim has to be a positive integer."))
new(
dim,
fill(ZZ(0), dim),
- (n::Int, k::Int, f::Function) -> gen(n, k, f)::fmpz,
+ (n::Int, k::Int, f::Function) -> gen(n, k, f)::ℤInt,
)
end
end
@@ -364,11 +767,11 @@ function Base.iterate(T::RecTriangle, n)
end
Base.length(R::RecTriangle) = R.dim
-Base.eltype(R::RecTriangle) = fmpz
+Base.eltype(R::RecTriangle) = ℤInt
function PolyArray(T::ℤTri)
P = Polynomial(T)
- # P = ReversedPolynomial(T)
+ # P = ReversePolynomial(T)
dim = length(T)
U = ZTri(dim)
for n = 1:dim
@@ -398,6 +801,32 @@ end
PolyVal2(T::ℤTri) = PolyValue(T, 2)
PolyVal3(T::ℤTri) = PolyValue(T, 3)
+const TRAITS = Function[
+ Flat,
+ Reverse,
+ Inverse,
+ RevInv,
+ InvRev,
+ sum,
+ EvenSum,
+ OddSum,
+ AltSum,
+ DiagSum,
+ Middle,
+ Central,
+ LeftSide,
+ RightSide,
+ PosHalf,
+ NegHalf,
+ PolyVal2,
+ PolyVal3,
+ TransUnos,
+ TransAlts,
+ TransSqrs,
+ TransNat0,
+ TransNat1
+ ]
+
# START-TEST-########################################################
# using Test
@@ -462,7 +891,7 @@ function demo()
DiagonalTriangle(Catalan(9)) |> println
P = Polynomial(T)
- p = P[3]
+ p = P[3]
p |> println
typeof(p) |> println
s = subst(p, 2)
@@ -470,8 +899,7 @@ function demo()
s = subst(p, QQ(1, 3))
s |> println
- T = Telescope(6, n -> ZZ(n))
- println(T)
+ T = Telescope(6, n -> ZZ(n))
println(isa(collect(T), ℤTri))
end
diff --git a/src/TrianglesExamples.jl b/src/TrianglesExamples.jl
index 96aa77d..2962ca8 100644
--- a/src/TrianglesExamples.jl
+++ b/src/TrianglesExamples.jl
@@ -10,13 +10,14 @@ using Nemo, TrianglesBase
export LahNumbers, LahTriangle, LahTransform, SchröderBTriangle, SchröderLTriangle, MotzkinTriangle
export Catalan, CatalanTriangle, CatalanTransform, CatalanBallot, ExtCatalanTriangle
export BernoulliPolynomial, PascalTriangle, SchroederBigTriangle
-export EulerianTriangle, NarayanaTriangle, NarayanaTransform
+export EulerianTriangle, EulerianTriangle2, NarayanaTriangle, NarayanaTransform
export EulerianTransform, MotzkinTransform, SchroederBigTransform
export JacobsthalTriangle, JacobsthalTransform, FibonacciTriangle, FibonacciTransform
export StirlingSetTriangle, StirlingCycleTriangle
export StirlingSetTransform, StirlingCycleTransform
export I132393, I048993, I271703, I094587, I008279, I225478, T132393, T048993
export T094587, T008279, T225478, T271703
+export TRIANGLES
function PrimeDivisors(n)
n < 2 && return ZInt[]
@@ -36,7 +37,7 @@ function LahNumbers(n::Int64)
CacheLah[n] = row
end
-LahNumbers(n, k) = LahNumbers(n + 1)[k + 1]
+LahNumbers(n, k) = LahNumbers(n)[k + 1]
function LahTriangle(size)
length(CacheLah) < size && LahNumbers(size)
@@ -133,6 +134,7 @@ function R094587(n::Int, k::Int, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
(n - k) * prevrow(k) + prevrow(k - 1)
end
+
"""
Recurrence for A008279. Number of permutations of n things k at a time.
(Falling factorials)
@@ -143,7 +145,7 @@ function R008279(n::Int, k::Int, prevrow::Function)
end
"""
-Iterates over the first n rows of `A132393`.
+xIterates over the first n rows of `A132393`.
Triangle of unsigned Stirling numbers of the first kind.
"""
I132393(n) = RecTriangle(n, R132393)
@@ -179,12 +181,18 @@ Iterates over the first n rows of `A094587`.
"""
I094587(n) = RecTriangle(n, R094587)
T094587(dim) = ZTri(I094587(dim))
+# T094587(dim) = Reverse(ZTri(I008279(dim)))
+
+
"""
Iterates over the first n rows of `A008279`.
"""
I008279(n) = RecTriangle(n, R008279)
T008279(dim) = ZTri(I008279(dim))
+FallFactTriangle(dim) = ZTri(I008279(dim))
+PermCoeffsTriangle(dim) = ZTri(I008279(dim))
+
function R225478(n, k, prevrow::Function)
(k == 0 && n == 0) && return ZZ(1)
4 * prevrow(k - 1) + (4 * n - 1) * prevrow(k)
@@ -255,6 +263,25 @@ EulerianTransform(A::ℤSeq) = Eulerian.(Telescope(A))
# (PARI) t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
#########################################################
+const CacheEulerian2 = Dict{Tuple{Int,Int},fmpz}()
+function EulerianNumbers2(n, k)
+ haskey(CacheEulerian2, (n, k)) && return CacheEulerian2[(n, k)]
+ CacheEulerian2[(n, k)] = if (k == n)
+ ZZ(1)
+ elseif (k <= 0) || (k > n)
+ ZZ(0)
+ else
+ (n - k + 1) * EulerianNumbers2(n - 1, k - 1) +
+ (k) * EulerianNumbers2(n - 1, k)
+ end
+end
+
+EulerianTriangle2(dim) = [[EulerianNumbers2(n, k) for k = 0:n] for n = 0:dim - 1]
+Eulerian2(n) = EulerianTriangle2(n + 1)[n + 1]
+Eulerian2(n, k) = Eulerian2(n)[k + 1]
+Eulerian2(A::ℤSeq) = LinMap(Eulerian2, A, length(A))
+EulerianTransform2(A::ℤSeq) = Eulerian2.(Telescope(A))
+
const CacheNarayana = Dict{Tuple{Int,Int},fmpz}()
function NarayanaNumbers(n::Int, k::Int)
haskey(CacheNarayana, (n, k)) && return CacheNarayana[(n, k)]
@@ -341,6 +368,23 @@ function transforms(trans)
trans(Nut) |> Println
end
+const TRIANGLES = Function[
+ BinomialTriangle,
+ CatalanTriangle,
+ EulerianTriangle,
+ FibonacciTriangle,
+ LaguerreTriangle,
+ LahTriangle,
+ MotzkinTriangle,
+ NarayanaTriangle,
+ SchröderBTriangle,
+ SchröderLTriangle,
+ StirlingCycleTriangle,
+ StirlingSetTriangle,
+ T008279
+ ]
+
+
# START-TEST-########################################################
using Test
@@ -397,8 +441,11 @@ function demo()
transforms(LaguerreTransform)
Println.(EulerianTriangle(8))
+ Println.(EulerianTriangle2(8))
transforms(EulerianTransform)
+ Println.(Inverse(EulerianTriangle2(8)))
+
Println.(NarayanaTriangle(8))
transforms(NarayanaTransform)
@@ -414,12 +461,13 @@ end
function main()
test()
- #demo()
+ demo()
perf()
end
-main()
-
+#main()
+Println.(Reverse(DiagonalTriangle(FallFactTriangle(12))))
+Println.(Reverse(FallFactTriangle(12)))
end # module
diff --git a/src/TrianglesExplorer.jl b/src/TrianglesExplorer.jl
index fee5863..652d695 100644
--- a/src/TrianglesExplorer.jl
+++ b/src/TrianglesExplorer.jl
@@ -15,34 +15,38 @@ const WARNING_ON_NOTFOUND = false
const Kind = ["Std", "Rev", "Inv", "RevInv", "InvRev"]
const Triangles = LittleDict{String, Function}(
- "Binomial" => BinomialTriangle,
- "Catalan" => CatalanTriangle,
- "Eulerian" => EulerianTriangle,
- "Fibonacci" => FibonacciTriangle,
- "Laguerre" => LaguerreTriangle,
- "Lah" => LahTriangle,
- "Motzkin" => MotzkinTriangle,
- "Narayana" => NarayanaTriangle,
- "SchroederB" => SchröderBTriangle,
+ "Binomial" => BinomialTriangle,
+ "Catalan" => CatalanTriangle,
+ "Eulerian" => EulerianTriangle,
+ "Fibonacci" => FibonacciTriangle,
+ "Laguerre" => LaguerreTriangle,
+ "Lah" => LahTriangle,
+ "Motzkin" => MotzkinTriangle,
+ "Narayana" => NarayanaTriangle,
+ "SchroederB" => SchröderBTriangle,
"SchroederL" => SchröderLTriangle,
- "StirlingCycle" => StirlingCycleTriangle,
- "StirlingSet" => StirlingSetTriangle,
- "PermCoeffs" => T008279
+ "StirlingCycle" => StirlingCycleTriangle,
+ "StirlingSet" => StirlingSetTriangle,
+ "PermCoeffs" => T008279
)
const Traits = LittleDict{String, Function}(
- "Triangle" => Flat,
- "Sum" => sum,
- "EvenSum" => EvenSum,
- "OddSum" => OddSum,
+ "Triangle" => Flat,
+ "Reverse" => Reverse,
+ "Inverse" => Inverse,
+ "RevInv" => RevInv,
+ "InvRev" => InvRev,
+ "Sum" => sum,
+ "EvenSum" => EvenSum,
+ "OddSum" => OddSum,
"AltSum" => AltSum,
- "DiagSum" => DiagSum,
- "Middle" => Middle,
- "Central" => Central,
- "LeftSide" => LeftSide,
- "RightSide" => RightSide,
- "PosHalf" => PosHalf,
- "NegHalf" => NegHalf,
+ "DiagSum" => DiagSum,
+ "Middle" => Middle,
+ "Central" => Central,
+ "LeftSide" => LeftSide,
+ "RightSide" => RightSide,
+ "PosHalf" => PosHalf,
+ "NegHalf" => NegHalf,
"PolyVal2" => PolyVal2,
"PolyVal3" => PolyVal3,
"TransUnos" => TransUnos,
@@ -64,24 +68,24 @@ function Show(io, name, kind, trait, seq, savetofile=false)
end
end
-function TriangleVariant(Tri, dim, kind="Std")
+function TriangleVariant(Tri, dim, kind="Std")
if ! (kind in Kind)
@warn("No valid kind!")
return []
end
-
+
M = Tri(dim)
kind == "Std" && return M
- kind == "Rev" && return reverse.(M)
+ kind == "Rev" && return reverse.(M)
kind == "InvRev" && (M = reverse.(M))
invM = InverseTriangle(M)
- (kind == "Inv" || kind == "InvRev"
+ (kind == "Inv" || kind == "InvRev"
|| invM == []) && return invM
- return reverse.(invM)
+ return reverse.(invM)
end
function Explore(triangle, kind, trait, dim)
- T = TriangleVariant(Triangles[triangle], dim, kind)
+ T = TriangleVariant(Triangles[triangle], dim, kind)
seq = Traits[trait](T)
Show(stdout, triangle, kind, trait, seq)
end
@@ -90,7 +94,7 @@ const LEN = 32
function Explore(triangle, kind, trait)
dim = 32
- T = TriangleVariant(Triangles[triangle], dim, kind)
+ T = TriangleVariant(Triangles[triangle], dim, kind)
seq = Traits[trait](T)
anum = GetSeqnum(seq)
anum === nothing && (anum = "nothing")
@@ -102,25 +106,25 @@ function Explore(trait, dim)
for (name, triangle) in Triangles
for kind in Kind
T = TriangleVariant(triangle, dim, kind)
- if T != []
+ if T != []
seq = Traits[trait](T)
Show(stdout, name, kind, trait, seq)
- end
+ end
end
end
end
# The BIG LIST goes to data/profile.txt.
-function Explore(savetofile=false)
+function Explore(savetofile::Bool)
@warn "This will take several minutes and produce the file 'profile.txt' in the data directory."
open(profilepath(), "a") do io
for (name, triangle) in Triangles
for kind in Kind
- T = TriangleVariant(triangle, LEN, kind)
+ T = TriangleVariant(triangle, LEN, kind)
if T != []
for (trait, f) in Traits
- Show(io, name, kind, trait, f(T), savetofile)
+ Show(io, name, kind, trait, f(T), savetofile)
end
end
end
@@ -150,11 +154,11 @@ function demo()
end
function perf()
- Explore(true)
+ Explore(true)
end
function main()
- test()
+ #test()
demo()
#perf()
end
diff --git a/src/TrianglesTraitCard.jl b/src/TrianglesTraitCard.jl
index 31e71cc..3bf1150 100644
--- a/src/TrianglesTraitCard.jl
+++ b/src/TrianglesTraitCard.jl
@@ -5,9 +5,11 @@
module TriangleTraitCard
-using Nemo, TrianglesBase
+using Nemo, TrianglesBase, TrianglesExamples, TrianglesUtils
-export TraitCard
+export TraitCard, xTraitCard, TransTraits, AssociatedTriangles
+
+const SEARCH = true
function TraitCard(T::ℚTri, name)
@@ -16,31 +18,131 @@ function TraitCard(T::ℚTri, name)
println()
end
-function TraitCard(T::ℤTri, name, N)
+function TraitCard(T::ℤTri, name, N, an=false)
+ an = an && ! oeis_notinstalled()
+
println("\n=================")
println(name)
println()
P = Polynomial(T)
- PA = PolyArray(T)
len = max(N, min(N - 1, length(T)))
+ len = len < 10 ? len : 10
+
+ S = Flat(T); print("Triangle: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
- S = Flat(T); print("Triangle: "); S[1:len] |> Println
- S = sum(T); print("Sum: "); S[1:len] |> Println
- S = EvenSum(T); print("EvenSum: "); S[1:len] |> Println
- S = OddSum(T); print("OddSum: "); S[1:len] |> Println
- S = AltSum(T); print("AltSum: "); S[1:len] |> Println
- S = DiagSum(T); print("DiagSum: "); S[1:len] |> Println
- S = Central(T); print("Central: "); S[1:min(len, length(S))] |> Println
- S = LeftSide(T); print("LeftSide: "); S[1:len] |> Println
- S = RightSide(T); print("RightSide: "); S[1:len] |> Println
- S = PosHalf(P); print("PosHalf: "); S[1:len] |> Println
- S = NegHalf(P); print("NegHalf: "); S[1:len] |> Println
- S = PolyValue(PA, 2); print("PolyVal2: "); S[1:len] |> Println
- S = PolyValue(PA, 3); print("PolyVal3: "); S[1:len] |> Println
- S = Trans(T, [ZZ(1) for n = 0:32]); print("TransUnos: "); S[1:len] |> Println
- S = Trans(T, [ZZ((-1)^n) for n = 0:32]); print("TransAlts: "); S[1:len] |> Println
- S = Trans(T, [ZZ(n) for n = 0:32]); print("TransNat0: "); S[1:len] |> Println
- S = Trans(T, [ZZ(n) for n = 1:32]); print("TransNat1: "); S[1:len] |> Println
+ R = Reverse(T)
+ FR = Flat(R); print("Reverse: ")
+ an && print(GetSeqnum(FR, SEARCH), " ")
+ FR[1:len] |> Println
+
+ I = Inverse(T)
+ if I != []
+ FI = Flat(I); print("Inverse: ")
+ an && print(GetSeqnum(FI, SEARCH), " ")
+ FI[1:len] |> Println
+
+ RI = Reverse(I)
+ RI = Flat(RI); print("RevInv: ")
+ an && print(GetSeqnum(RI, SEARCH), " ")
+ RI[1:len] |> Println
+ end
+
+ IR = Inverse(R)
+ if IR != []
+ IR = Flat(IR); print("InvRev: ")
+ an && print(GetSeqnum(IR, SEARCH), " ")
+ IR[1:len] |> Println
+ end
+
+ S = DiagonalTriangle(T)
+ S = Flat(S);
+ print("Diagonal: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+
+ S = sum(T); print("Sum: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = EvenSum(T); print("EvenSum: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = OddSum(T); print("OddSum: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = AltSum(T); print("AltSum: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = DiagSum(T); print("DiagSum: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Central(T); print("Central: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:min(len, length(S))] |> Println
+ S = LeftSide(T); print("LeftSide: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = RightSide(T); print("RightSide: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = PosHalf(P); print("PosHalf: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = NegHalf(P); print("NegHalf: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Trans(T, [ZZ(1) for n = 0:32]); print("TransUnos: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Trans(T, [ZZ((-1)^n) for n = 0:32]); print("TransAlts: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Trans(T, [ZZ(n^2) for n = 0:32]); print("TransSqrs: ");
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Trans(T, [ZZ(n) for n = 0:32]); print("TransNat0: ");
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = Trans(T, [ZZ(n) for n = 1:32]); print("TransNat1: ");
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = PolyVal2(T); print("PolyVal2: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+ S = PolyVal3(T); print("PolyVal3: ")
+ an && print(GetSeqnum(S, SEARCH), " ")
+ S[1:len] |> Println
+end
+
+"""
+Print the standard traits generated by linear transformation.
+```
+julia> TransTraits(BinomialTriangle)
+BinomialTriangle
+TransUnos: A000079 [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]
+TransAlts: A000007 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+TransSqrs: A001788 [0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520]
+TransNat0: A001787 [0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304]
+TransNat1: A001792 [1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816]
+```
+"""
+function TransTraits(triangle::Function)
+ dim = 32
+ T = triangle(dim)
+ println("$triangle ")
+ len = min(10, min(dim - 1, length(T)))
+
+ S = Trans(T, [ZZ(1) for n = 0:dim]); anum = GetSeqnum(S, false)
+ print("TransUnos: ", anum, " "); S[1:len] |> Println
+ S = Trans(T, [ZZ((-1)^n) for n = 0:dim]); anum = GetSeqnum(S, false)
+ print("TransAlts: ", anum, " "); S[1:len] |> Println
+ S = Trans(T, [ZZ(n^2) for n = 0:dim]); anum = GetSeqnum(S, false)
+ print("TransSqrs: ", anum, " "); S[1:len] |> Println
+ S = Trans(T, [ZZ(n) for n = 0:dim]); anum = GetSeqnum(S, false)
+ print("TransNat0: ", anum, " "); S[1:len] |> Println
+ S = Trans(T, [ZZ(n) for n = 1:dim]); anum = GetSeqnum(S, false)
+ print("TransNat1: ", anum, " "); S[1:len] |> Println
println()
end
@@ -76,15 +178,54 @@ function AssociatedTriangles(T::ℤTri, N)
Println.(S[1:len])
println()
S = Flat(S); print("PolyVal: "); S[1:len] |> Println
- println()
+
end
-function TraitCard(triangle::Function, dim=8)
+"""
+Prints a list of traits of the triangle without Anums
+and, if assoc=true, also a list of associated triangles.
+"""
+function TraitCard(triangle::Function, dim=8, assoc=false)
T = triangle(dim)
TraitCard(T, "$triangle ", dim)
AssociatedTriangles(T, dim)
end
+"""
+Prints a list of traits of the triangle with Anums.
+xTraitCard is slower and uses much more resources than TraitCard.
+```
+julia> xTraitCard(LaguerreTriangle)
+Triangle: A021009 [1, 1, 1, 2, 4, 1, 6, 18, 9, 1]
+Reverse: A021010 [1, 1, 1, 1, 4, 2, 1, 9, 18, 6]
+Inverse: A021009 [1, -1, 1, 2, -4, 1, -6, 18, -9, 1]
+RevInv: A021010 [1, 1, -1, 1, -4, 2, 1, -9, 18, -6]
+Diagonal: A084950 [1, 1, 2, 1, 6, 4, 24, 18, 1, 120]
+Sum: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+EvenSum: A331325 [1, 1, 3, 15, 97, 745, 6571, 65359, 723969, 8842257]
+OddSum: A331326 [0, 1, 4, 19, 112, 801, 6756, 65563, 717760, 8729857]
+AltSum: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+DiagSum: C001040 [1, 1, 3, 10, 43, 225, 1393, 9976, 81201, 740785]
+Central: A295383 [1, 4, 72, 2400, 117600, 7620480, 614718720, 59364264960]
+LeftSide: A000142 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
+RightSide: A000012 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
+PosHalf: A025167 [1, 3, 17, 139, 1473, 19091, 291793, 5129307, 101817089]
+NegHalf: A025166 [1, -1, 1, 7, -127, 1711, -23231, 334391, -5144063, 84149983]
+TransUnos: A002720 [1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114]
+TransAlts: A009940 [1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400]
+TransSqrs: A105219 [0, 1, 8, 63, 544, 5225, 55656, 653023, 8379008, 116780049]
+TransNat0: A103194 [0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977]
+TransNat1: C000262 [1, 3, 13, 73, 501, 4051, 37633, 394353, 4596553, 58941091]
+PolyVal2: A087912 [1, 3, 14, 86, 648, 5752, 58576, 671568, 8546432, 119401856]
+PolyVal3: A277382 [1, 4, 23, 168, 1473, 14988, 173007, 2228544, 31636449]
+```
+"""
+function xTraitCard(triangle::Function)
+ dim = 32
+ T = triangle(dim)
+ TraitCard(T, "$triangle ", dim, true)
+end
+
# START-TEST-########################################################
using TrianglesExamples
@@ -100,18 +241,38 @@ function test()
TraitCard(JacobsthalTriangle, dim)
TraitCard(FibonacciTriangle, dim)
TraitCard(EulerianTriangle, dim)
+
+ TransTraits(BinomialTriangle)
end
function demo()
+ xTraitCard(BinomialTriangle)
+ xTraitCard(LaguerreTriangle)
+ xTraitCard(LahTriangle)
+ xTraitCard(CatalanTriangle)
+ xTraitCard(MotzkinTriangle)
+ xTraitCard(NarayanaTriangle)
+ xTraitCard(SchroederBigTriangle)
+ xTraitCard(JacobsthalTriangle)
+ xTraitCard(FibonacciTriangle)
+ xTraitCard(EulerianTriangle)
+ xTraitCard(EulerianTriangle2)
end
function perf()
+ for t in TRIANGLES
+ xTraitCard(t)
+ end
end
function main()
- test()
- demo()
- perf()
+ #test()
+ #demo()
+ #perf()
+ #xTraitCard(EulerianTriangle)
+ #xTraitCard(EulerianTriangle2)
+ # xTraitCard(MotzkinTriangle)
+ xTraitCard(T094587)
end
main()
diff --git a/src/TrianglesUtils.jl b/src/TrianglesUtils.jl
index bda081e..a54c33c 100644
--- a/src/TrianglesUtils.jl
+++ b/src/TrianglesUtils.jl
@@ -6,7 +6,7 @@
module TrianglesUtils
using Nemo, TrianglesBase, HTTP
-export Show, GetSeqnum, SeqToString, oeis_search, search_failed
+export Show, GetSeqnum, SeqToString, oeis_search, oeis_notinstalled, search_failed
export profilepath
const srcdir = realpath(joinpath(dirname(@__FILE__)))
@@ -90,13 +90,13 @@ function SeqToString(seq::ℤSeq)
end
# increases accuracy and prevents premature matches
-const minlen = 30 # fragil!
+const minlen = 30 # fragil! do not reduce!
function GetSeqnum(seq::ℤSeq)
str = SeqToString(seq)
soff = 1; loff = 10
- println(seq)
- println(str)
+ #println(seq)
+ #println(str)
for ln ∈ eachline(oeis_file())
ln[1] == '#' && continue
l = replace(ln, "-" => "")
diff --git a/test/perftests.jl b/test/perftests.jl
index 9730731..0770391 100644
--- a/test/perftests.jl
+++ b/test/perftests.jl
@@ -1,8 +1,8 @@
# This file is part of IntegerTriangles.
# Copyright Peter Luschny. License is MIT.
-# Version of: UTC 2021-04-14 11:04:40
-# 72326c02-9d00-11eb-2152-a5f1649da950
+# Version of: UTC 2021-04-19 09:45:06
+# 28a626d0-a0e3-11eb-184e-9f8c20e6c3f3
# Do not edit this file, it is generated from the modules and will be overwritten!
# Edit the modules in the src directory and build this file with BuildTriangles.jl!
@@ -18,10 +18,13 @@ start = Dates.now()
# +++ TrianglesBase.jl +++
# +++ TrianglesExamples.jl +++
# +++ TrianglesExplorer.jl +++
-Explore(true)
+Explore(true)
# +++ TrianglesPlot.jl +++
# +++ TrianglesTables.jl +++
# +++ TrianglesTraitCard.jl +++
+for t in TRIANGLES
+xTraitCard(t)
+end
# +++ TrianglesUtils.jl +++
GetSeqnum(ℤInt[1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10,
12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51,
diff --git a/test/runtests.jl b/test/runtests.jl
index cc4ef5f..dce17e1 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1,8 +1,8 @@
# This file is part of IntegerTriangles.
# Copyright Peter Luschny. License is MIT.
-# Version of: UTC 2021-04-14 11:04:40
-# 723244f0-9d00-11eb-175c-7330d2cf4b47
+# Version of: UTC 2021-04-19 09:45:06
+# 28a5b1a0-a0e3-11eb-0d61-c3f98c353107
# Do not edit this file, it is generated from the modules and will be overwritten!
# Edit the modules in the src directory and build this file with BuildTriangles.jl!
@@ -70,6 +70,7 @@ TraitCard(SchroederBigTriangle, dim)
TraitCard(JacobsthalTriangle, dim)
TraitCard(FibonacciTriangle, dim)
TraitCard(EulerianTriangle, dim)
+TransTraits(BinomialTriangle)
# *** TrianglesUtils.jl *********
T = LaguerreTriangle(7)
println("\nThe $Laguerre triangle in different formats:")