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begins:\n  0: 1;\n  1: 1,   2;\n  2: 1,   8,     4;\n  3: 1,  20,    52,       8;\n  4: 1,  40,   292,     320,       16;\n  5: 1,  70,  1092,    3824,     1936,        32;\n  6: 1, 112,  3192,   25664,    47824,     11648,        64;\n  7: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;\n  8: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;\n  9: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n^2+n)*x)*P(n-1,x) - 2*n*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n^2 + n)*x)*P[n - 1, x] - 2*n*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {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] = (1 + (n^2 + n)*x)*v[i - 1] - 2*n*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A071951","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   2,       1;\n   3:   4,       8,        1;\n   4:   8,      52,       20,         1;\n   5:  16,     320,      292,        40,         1;\n   6:  32,    1936,     3824,      1092,        70,        1;\n   7:  64,   11648,    47824,     25664,      3192,      112,       1;\n   8: 128,   69952,   585536,    561104,    121424,     7896,     168,     1;\n   9: 256,  419840,  7096384,  11807616,   4203824,   453056,   17304,   240,   1;\n  10: 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n^2-n)*x)*P(n-1,x) - (2*n-2)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n^2 - n)*x)*P[n - 1, x] - (2*n - 2)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - n)*x)*v[i - 1] - (2*n - 2)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A191935","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   2;\n   3: 1,   8,     4;\n   4: 1,  20,    52,       8;\n   5: 1,  40,   292,     320,       16;\n   6: 1,  70,  1092,    3824,     1936,        32;\n   7: 1, 112,  3192,   25664,    47824,     11648,        64;\n   8: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;\n   9: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;\n  10: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n^2-n)*x)*P(n-1,x) - (2*n-2)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n^2 - n)*x)*P[n - 1, x] - (2*n - 2)*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - n)*x)*v[i - 1] - (2*n - 2)*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-1dda68ade0f78df3","steps":[],"edges":[],"text":"A191935"},"headline":"A191935: Triangle read by rows of Legendre-Stirling numbers of the second kind.","chain":null}