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v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A019538","target":"triangle","first_row":2,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    2;\n   3: 1,    6,     6;\n   4: 1,   14,    36,     24;\n   5: 1,   30,   150,    240,     120;\n   6: 1,   62,   540,   1560,    1800,      720;\n   7: 1,  126,  1806,   8400,   16800,    15120,     5040;\n   8: 1,  254,  5796,  40824,  126000,   191520,   141120,    40320;\n   9: 1,  510, 18150, 186480,  834120,  1905120,  2328480,  1451520,   362880;\n  10: 1, 1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000, 16329600, 3628800;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A090582","target":"reversed_triangle","first_row":2,"oeis_offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       2,        1;\n   3:       6,        6,        1;\n   4:      24,       36,       14,        1;\n   5:     120,      240,      150,       30,        1;\n   6:     720,     1800,     1560,      540,       62,       1;\n   7:    5040,    15120,    16800,     8400,     1806,     126,      1;\n   8:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;\n   9:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;\n  10: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A131689","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,   2;\n  3: 0, 1,   6,     6;\n  4: 0, 1,  14,    36,     24;\n  5: 0, 1,  30,   150,    240,    120;\n  6: 0, 1,  62,   540,   1560,   1800,     720;\n  7: 0, 1, 126,  1806,   8400,  16800,   15120,    5040;\n  8: 0, 1, 254,  5796,  40824, 126000,  191520,  141120,   40320;\n  9: 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-2cabfd060b04f15d","steps":[],"edges":[],"text":"A131689"},"headline":"A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","chain":null}