Dev0.3

.gitignore

@@ -1,5 +1,6 @@
 Manifest.toml
 local/*
+profiles/*
 docs/build/*
 .vscode/*
 .github/*

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).
 

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
+}

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
+}

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
+}

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
+}

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
+}

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
+}

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
+}

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"
+		}
+	]
+}

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
+}

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"
+		}
+	]
+}