Skip to content

Latest commit

 

History

History
246 lines (212 loc) · 17 KB

Eulerian2.tbl.md

File metadata and controls

246 lines (212 loc) · 17 KB
Raw

Eulerian2

['A340556', 'A008517', 'A112007', 'A163936']

Eulerian2 Triangle view

Row Seq
Row0 [1]
Row1 [0, 1]
Row2 [0, 1, 2]
Row3 [0, 1, 8, 6]
Row4 [0, 1, 22, 58, 24]
Row5 [0, 1, 52, 328, 444, 120]
Row6 [0, 1, 114, 1452, 4400, 3708, 720]
Row7 [0, 1, 240, 5610, 32120, 58140, 33984, 5040]
Row8 [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320]
Row9 [0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880]

Eulerian2 Triangles

Flat Seq
Triangle [[1], [0, 1], [0, 1, 2], [0, 1, 8, 6], [0, 1, 22, 58, 24], [0, 1, 52, 328, 444, 120], [0, 1, 114, 1452, 4400, 3708, 720], [0, 1, 240, 5610, 32120, 58140, 33984, 5040], [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320], [0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880]]
RevTabl [[1], [1, 0], [2, 1, 0], [6, 8, 1, 0], [24, 58, 22, 1, 0], [120, 444, 328, 52, 1, 0], [720, 3708, 4400, 1452, 114, 1, 0], [5040, 33984, 58140, 32120, 5610, 240, 1, 0], [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0], [362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1, 0]]
AntiDiag [[1], [0], [0, 1], [0, 1], [0, 1, 2], [0, 1, 8], [0, 1, 22, 6], [0, 1, 52, 58], [0, 1, 114, 328, 24], [0, 1, 240, 1452, 444]]
AccTabl [[1], [0, 1], [0, 1, 3], [0, 1, 9, 15], [0, 1, 23, 81, 105], [0, 1, 53, 381, 825, 945], [0, 1, 115, 1567, 5967, 9675, 10395], [0, 1, 241, 5851, 37971, 96111, 130095, 135135], [0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025], [0, 1, 1005, 68265, 1130765, 6896265, 19336329, 30362625, 34096545, 34459425]]
RevAccTabl [[1], [1, 0], [3, 1, 0], [15, 9, 1, 0], [105, 81, 23, 1, 0], [945, 825, 381, 53, 1, 0], [10395, 9675, 5967, 1567, 115, 1, 0], [135135, 130095, 96111, 37971, 5851, 241, 1, 0], [2027025, 1986705, 1645569, 860265, 216245, 20445, 495, 1, 0], [34459425, 34096545, 30362625, 19336329, 6896265, 1130765, 68265, 1005, 1, 0]]
AccRevTabl [[1], [1, 1], [2, 3, 3], [6, 14, 15, 15], [24, 82, 104, 105, 105], [120, 564, 892, 944, 945, 945], [720, 4428, 8828, 10280, 10394, 10395, 10395], [5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135], [40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025], [362880, 4096800, 15123096, 27563160, 33328660, 34391160, 34458420, 34459424, 34459425, 34459425]]
DiffxTabl [[1], [0, 2], [0, 2, 6], [0, 2, 24, 24], [0, 2, 66, 232, 120], [0, 2, 156, 1312, 2220, 720], [0, 2, 342, 5808, 22000, 22248, 5040], [0, 2, 720, 22440, 160600, 348840, 237888, 40320], [0, 2, 1482, 79800, 979000, 3864120, 5497128, 2729088, 362880], [0, 2, 3012, 269040, 5312500, 34593000, 87080448, 88210368, 33605280, 3628800]]

Eulerian2 Row sums

Trait Seq
RowSum [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
EvenSum [1, 0, 2, 8, 46, 496, 5234, 66344, 1021918, 17237488]
OddSum [0, 1, 1, 7, 59, 449, 5161, 68791, 1005107, 17221937]
AltSum [1, -1, 1, 1, -13, 47, 73, -2447, 16811, 15551]
AbsSum [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
AccSum [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750, 126351225]
AccRevSum [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500, 252702450]
DiagSum [1, 0, 1, 1, 3, 9, 29, 111, 467, 2137]

Eulerian2 Transforms

Trans Seq
RowLcm [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600, 8842151410302701559591432000000]
RowGcd [1, 1, 2, 2, 2, 4, 2, 2, 2, 4]
RowMax [1, 1, 2, 8, 58, 444, 4400, 58140, 785304, 12440064]
CentralE [1, 1, 22, 1452, 195800]
CentralO [0, 1, 52, 5610, 1062500]
ColMiddle [1, 0, 1, 1, 22, 52, 1452, 5610, 195800, 1062500]
ColLeft [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
ColRight [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
BinConv [1, 1, 4, 33, 392, 6145, 119724, 2789465, 75660080, 2341894185]
InvBinConv [1, 1, 0, -15, -80, 665, 17136, 55209, -4162752, -83069415]
TransSqrs [0, 1, 9, 87, 995, 13265, 202545, 3489255, 66981915, 1418241825]
TransNat0 [0, 1, 5, 35, 315, 3465, 45045, 675675, 11486475, 218243025]
TransNat1 [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500, 252702450]
PosHalf [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912, 282137824]
NegHalf [1, 1, 0, -6, -12, 144, 1080, -5184, -127008, -95904]

Eulerian2 Diagonals as rows

DiagRow Seq
DiagRow0 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
DiagRow1 [0, 1, 8, 58, 444, 3708, 33984, 341136, 3733920, 44339040]
DiagRow2 [0, 1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216]
DiagRow3 [0, 1, 52, 1452, 32120, 644020, 12440064, 238904904, 4642163952, 92199790224]
DiagRow4 [0, 1, 114, 5610, 195800, 5765500, 155357384, 4002695088, 101180433024, 2549865473424]
DiagRow5 [0, 1, 240, 19950, 1062500, 44765000, 1648384304, 56041398784, 1818188642304, 57494373464592]
DiagRow6 [0, 1, 494, 67260, 5326160, 314369720, 15548960784, 687720046384, 28299910066112, 1111747472569680]
DiagRow7 [0, 1, 1004, 218848, 25243904, 2051482776, 134323420224, 7634832149392, 394365587815520, 19076135772884080]
DiagRow8 [0, 1, 2026, 695038, 114876376, 12669817776, 1084676512416, 78391384831312, 5036317938475648, 297593170417847920]
DiagRow9 [0, 1, 4072, 2170626, 507259276, 75016052228, 8308444327968, 756450802018384, 59958264360283168, 4297211671488276816]

Eulerian2 Diagonals as columns

DiagCol Seq
DiagCol0 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
DiagCol1 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
DiagCol2 [2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072]
DiagCol3 [6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626]
DiagCol4 [24, 444, 4400, 32120, 195800, 1062500, 5326160, 25243904, 114876376, 507259276]
DiagCol5 [120, 3708, 58140, 644020, 5765500, 44765000, 314369720, 2051482776, 12669817776, 75016052228]
DiagCol6 [720, 33984, 785304, 12440064, 155357384, 1648384304, 15548960784, 134323420224, 1084676512416, 8308444327968]
DiagCol7 [5040, 341136, 11026296, 238904904, 4002695088, 56041398784, 687720046384, 7634832149392, 78391384831312, 756450802018384]
DiagCol8 [40320, 3733920, 162186912, 4642163952, 101180433024, 1818188642304, 28299910066112, 394365587815520, 5036317938475648, 59958264360283168]
DiagCol9 [362880, 44339040, 2507481216, 92199790224, 2549865473424, 57494373464592, 1111747472569680, 19076135772884080, 297593170417847920, 4297211671488276816]

Eulerian2 Polynomial values as rows

PolyRow Seq
PolyRow0 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
PolyRow1 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
PolyRow2 [0, 3, 10, 21, 36, 55, 78, 105, 136, 171]
PolyRow3 [0, 15, 82, 237, 516, 955, 1590, 2457, 3592, 5031]
PolyRow4 [0, 105, 938, 3711, 10212, 22805, 44430, 78603, 129416, 201537]
PolyRow5 [0, 945, 13778, 74451, 258372, 694805, 1581270, 3197943, 5922056, 10242297]
PolyRow6 [0, 10395, 247210, 1822557, 7967268, 25771855, 68445870, 158095665, 329020552, 631527363]
PolyRow7 [0, 135135, 5240338, 52680189, 289894788, 1127219755, 3491379510, 9205055433, 21517930504, 45814612023]
PolyRow8 [0, 2027025, 128149802, 1755990327, 12159074916, 56809068605, 205123491150, 617068198467, 1619649615752, 3823957410921]
PolyRow9 [0, 34459425, 3551246162, 66313036227, 577622524356, 3241807557605, 13641831014550, 46812487574727, 137927928754952, 361017218715561]

Eulerian2 Polynomial values as columns

PolyCol Seq
PolyCol0 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
PolyCol1 [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
PolyCol2 [1, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162]
PolyCol3 [1, 3, 21, 237, 3711, 74451, 1822557, 52680189, 1755990327, 66313036227]
PolyCol4 [1, 4, 36, 516, 10212, 258372, 7967268, 289894788, 12159074916, 577622524356]
PolyCol5 [1, 5, 55, 955, 22805, 694805, 25771855, 1127219755, 56809068605, 3241807557605]
PolyCol6 [1, 6, 78, 1590, 44430, 1581270, 68445870, 3491379510, 205123491150, 13641831014550]
PolyCol7 [1, 7, 105, 2457, 78603, 3197943, 158095665, 9205055433, 617068198467, 46812487574727]
PolyCol8 [1, 8, 136, 3592, 129416, 5922056, 329020552, 21517930504, 1619649615752, 137927928754952]
PolyCol9 [1, 9, 171, 5031, 201537, 10242297, 631527363, 45814612023, 3823957410921, 361017218715561]

Eulerian2:Rev

[]

Eulerian2:Rev Triangle view

Row Seq
Row0 [1]
Row1 [1, 0]
Row2 [2, 1, 0]
Row3 [6, 8, 1, 0]
Row4 [24, 58, 22, 1, 0]
Row5 [120, 444, 328, 52, 1, 0]
Row6 [720, 3708, 4400, 1452, 114, 1, 0]
Row7 [5040, 33984, 58140, 32120, 5610, 240, 1, 0]
Row8 [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0]
Row9 [362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1, 0]

Eulerian2:Rev Triangles

Flat Seq
Triangle [[1], [1, 0], [2, 1, 0], [6, 8, 1, 0], [24, 58, 22, 1, 0], [120, 444, 328, 52, 1, 0], [720, 3708, 4400, 1452, 114, 1, 0], [5040, 33984, 58140, 32120, 5610, 240, 1, 0], [40320, 341136, 785304, 644020, 195800, 19950, 494, 1, 0], [362880, 3733920, 11026296, 12440064, 5765500, 1062500, 67260, 1004, 1, 0]]
RevTabl [[1], [0, 1], [0, 1, 2], [0, 1, 8, 6], [0, 1, 22, 58, 24], [0, 1, 52, 328, 444, 120], [0, 1, 114, 1452, 4400, 3708, 720], [0, 1, 240, 5610, 32120, 58140, 33984, 5040], [0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320], [0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880]]
AntiDiag [[1], [1], [2, 0], [6, 1], [24, 8, 0], [120, 58, 1], [720, 444, 22, 0], [5040, 3708, 328, 1], [40320, 33984, 4400, 52, 0], [362880, 341136, 58140, 1452, 1]]
AccTabl [[1], [1, 1], [2, 3, 3], [6, 14, 15, 15], [24, 82, 104, 105, 105], [120, 564, 892, 944, 945, 945], [720, 4428, 8828, 10280, 10394, 10395, 10395], [5040, 39024, 97164, 129284, 134894, 135134, 135135, 135135], [40320, 381456, 1166760, 1810780, 2006580, 2026530, 2027024, 2027025, 2027025], [362880, 4096800, 15123096, 27563160, 33328660, 34391160, 34458420, 34459424, 34459425, 34459425]]
RevAccTabl [[1], [1, 1], [3, 3, 2], [15, 15, 14, 6], [105, 105, 104, 82, 24], [945, 945, 944, 892, 564, 120], [10395, 10395, 10394, 10280, 8828, 4428, 720], [135135, 135135, 135134, 134894, 129284, 97164, 39024, 5040], [2027025, 2027025, 2027024, 2026530, 2006580, 1810780, 1166760, 381456, 40320], [34459425, 34459425, 34459424, 34458420, 34391160, 33328660, 27563160, 15123096, 4096800, 362880]]
AccRevTabl [[1], [0, 1], [0, 1, 3], [0, 1, 9, 15], [0, 1, 23, 81, 105], [0, 1, 53, 381, 825, 945], [0, 1, 115, 1567, 5967, 9675, 10395], [0, 1, 241, 5851, 37971, 96111, 130095, 135135], [0, 1, 495, 20445, 216245, 860265, 1645569, 1986705, 2027025], [0, 1, 1005, 68265, 1130765, 6896265, 19336329, 30362625, 34096545, 34459425]]
DiffxTabl [[1], [1, 0], [2, 2, 0], [6, 16, 3, 0], [24, 116, 66, 4, 0], [120, 888, 984, 208, 5, 0], [720, 7416, 13200, 5808, 570, 6, 0], [5040, 67968, 174420, 128480, 28050, 1440, 7, 0], [40320, 682272, 2355912, 2576080, 979000, 119700, 3458, 8, 0], [362880, 7467840, 33078888, 49760256, 28827500, 6375000, 470820, 8032, 9, 0]]

Eulerian2:Rev Row sums

Trait Seq
RowSum [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
EvenSum [1, 1, 2, 7, 46, 449, 5234, 68791, 1021918, 17221937]
OddSum [0, 0, 1, 8, 59, 496, 5161, 66344, 1005107, 17237488]
AltSum [1, 1, 1, -1, -13, -47, 73, 2447, 16811, -15551]
AbsSum [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
AccSum [1, 2, 8, 50, 420, 4410, 55440, 810810, 13513500, 252702450]
AccRevSum [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750, 126351225]
DiagSum [1, 1, 2, 7, 32, 179, 1186, 9077, 78756, 763609]

Eulerian2:Rev Transforms

Trans Seq
RowLcm [1, 1, 2, 24, 7656, 2366520, 852469200, 308503863360, 240006609023406825600, 8842151410302701559591432000000]
RowGcd [1, 1, 2, 2, 2, 4, 2, 2, 2, 4]
RowMax [1, 1, 2, 8, 58, 444, 4400, 58140, 785304, 12440064]
CentralE [1, 1, 22, 1452, 195800]
CentralO [1, 8, 328, 32120, 5765500]
ColMiddle [1, 1, 1, 8, 22, 328, 1452, 32120, 195800, 5765500]
ColLeft [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
ColRight [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
BinConv [1, 1, 4, 33, 392, 6145, 119724, 2789465, 75660080, 2341894185]
InvBinConv [1, -1, 0, 15, -80, -665, 17136, -55209, -4162752, 83069415]
TransSqrs [0, 0, 1, 12, 155, 2240, 36225, 651420, 12927915, 281080800]
TransNat0 [0, 0, 1, 10, 105, 1260, 17325, 270270, 4729725, 91891800]
TransNat1 [1, 1, 4, 25, 210, 2205, 27720, 405405, 6756750, 126351225]
PosHalf [1, 2, 10, 82, 938, 13778, 247210, 5240338, 128149802, 3551246162]
NegHalf [1, -2, 6, -18, 6, 846, -13338, 139374, -717498, -13142898]

Eulerian2:Rev Diagonals as rows

DiagRow Seq
DiagRow0 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
DiagRow1 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
DiagRow2 [2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072]
DiagRow3 [6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626]
DiagRow4 [24, 444, 4400, 32120, 195800, 1062500, 5326160, 25243904, 114876376, 507259276]
DiagRow5 [120, 3708, 58140, 644020, 5765500, 44765000, 314369720, 2051482776, 12669817776, 75016052228]
DiagRow6 [720, 33984, 785304, 12440064, 155357384, 1648384304, 15548960784, 134323420224, 1084676512416, 8308444327968]
DiagRow7 [5040, 341136, 11026296, 238904904, 4002695088, 56041398784, 687720046384, 7634832149392, 78391384831312, 756450802018384]
DiagRow8 [40320, 3733920, 162186912, 4642163952, 101180433024, 1818188642304, 28299910066112, 394365587815520, 5036317938475648, 59958264360283168]
DiagRow9 [362880, 44339040, 2507481216, 92199790224, 2549865473424, 57494373464592, 1111747472569680, 19076135772884080, 297593170417847920, 4297211671488276816]

Eulerian2:Rev Diagonals as columns

DiagCol Seq
DiagCol0 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
DiagCol1 [0, 1, 8, 58, 444, 3708, 33984, 341136, 3733920, 44339040]
DiagCol2 [0, 1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216]
DiagCol3 [0, 1, 52, 1452, 32120, 644020, 12440064, 238904904, 4642163952, 92199790224]
DiagCol4 [0, 1, 114, 5610, 195800, 5765500, 155357384, 4002695088, 101180433024, 2549865473424]
DiagCol5 [0, 1, 240, 19950, 1062500, 44765000, 1648384304, 56041398784, 1818188642304, 57494373464592]
DiagCol6 [0, 1, 494, 67260, 5326160, 314369720, 15548960784, 687720046384, 28299910066112, 1111747472569680]
DiagCol7 [0, 1, 1004, 218848, 25243904, 2051482776, 134323420224, 7634832149392, 394365587815520, 19076135772884080]
DiagCol8 [0, 1, 2026, 695038, 114876376, 12669817776, 1084676512416, 78391384831312, 5036317938475648, 297593170417847920]
DiagCol9 [0, 1, 4072, 2170626, 507259276, 75016052228, 8308444327968, 756450802018384, 59958264360283168, 4297211671488276816]

Eulerian2:Rev Polynomial values as rows

PolyRow Seq
PolyRow0 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
PolyRow1 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
PolyRow2 [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
PolyRow3 [6, 15, 26, 39, 54, 71, 90, 111, 134, 159]
PolyRow4 [24, 105, 236, 423, 672, 989, 1380, 1851, 2408, 3057]
PolyRow5 [120, 945, 2752, 5889, 10728, 17665, 27120, 39537, 55384, 75153]
PolyRow6 [720, 10395, 39208, 100125, 209088, 385135, 650520, 1030833, 1555120, 2256003]
PolyRow7 [5040, 135135, 660032, 2010951, 4812912, 9915335, 18423360, 31729887, 51548336, 79946127]
PolyRow8 [40320, 2027025, 12818912, 46589967, 127780416, 294396725, 601682400, 1126164627, 1970081792, 3266196921]
PolyRow9 [362880, 34459425, 282137824, 1223110881, 3843863424, 9903213505, 22261672800, 45279807489, 85290841216, 151152782049]

Eulerian2:Rev Polynomial values as columns

PolyCol Seq
PolyCol0 [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
PolyCol1 [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]
PolyCol2 [1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912, 282137824]
PolyCol3 [1, 1, 5, 39, 423, 5889, 100125, 2010951, 46589967, 1223110881]
PolyCol4 [1, 1, 6, 54, 672, 10728, 209088, 4812912, 127780416, 3843863424]
PolyCol5 [1, 1, 7, 71, 989, 17665, 385135, 9915335, 294396725, 9903213505]
PolyCol6 [1, 1, 8, 90, 1380, 27120, 650520, 18423360, 601682400, 22261672800]
PolyCol7 [1, 1, 9, 111, 1851, 39537, 1030833, 31729887, 1126164627, 45279807489]
PolyCol8 [1, 1, 10, 134, 2408, 55384, 1555120, 51548336, 1970081792, 85290841216]
PolyCol9 [1, 1, 11, 159, 3057, 75153, 2256003, 79946127, 3266196921, 151152782049]