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begins:\n  0:   1;\n  1:   2,      2;\n  2:   4,     16,        4;\n  3:   8,     88,       88,         8;\n  4:  16,    416,     1056,       416,        16;\n  5:  32,   1824,     9664,      9664,      1824,        32;\n  6:  64,   7680,    76224,    154624,     76224,      7680,        64;\n  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;\n  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;\n  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*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[(2 + 2*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];\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] = (2 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], 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":"A257609","target":"triangle","first_row":0,"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:   2,      2;\n  2:   4,     16,        4;\n  3:   8,     88,       88,         8;\n  4:  16,    416,     1056,       416,        16;\n  5:  32,   1824,     9664,      9664,      1824,        32;\n  6:  64,   7680,    76224,    154624,     76224,      7680,        64;\n  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;\n  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;\n  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*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[(2 + 2*n*x)*P[n - 1, x] + (2*x - 2*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] = (2 + 2*n*x)*v[i - 1] + (2*x - 2*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A257609","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,      2;\n  2:   4,     16,        4;\n  3:   8,     88,       88,         8;\n  4:  16,    416,     1056,       416,        16;\n  5:  32,   1824,     9664,      9664,      1824,        32;\n  6:  64,   7680,    76224,    154624,     76224,      7680,        64;\n  7: 128,  31616,   549504,   1999232,   1999232,    549504,     31616,      128;\n  8: 256, 128512,  3739648,  22587904,  39984640,  22587904,   3739648,   128512,    256;\n  9: 512, 518656, 24494080, 233058304, 670901248, 670901248, 233058304, 24494080, 518656, 512;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*n*x)*P(n-1,x) + (2*x-2*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[(2 + 2*n*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];\nTable[Reverse[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] = (2 + 2*n*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-41809911e05bb4ba","steps":[],"edges":[],"text":"A257609"},"headline":"A257609: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","chain":null}