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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (9*n - 7)*x)*v[i - 1] + (9*x - 9*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":"A257619","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,         44,            4;\n  3:   8,        564,          564,             8;\n  4:  16,       6436,        22560,          6436,            16;\n  5:  32,      71404,       637844,        637844,         71404,            32;\n  6:  64,     786948,     15470232,      36994952,      15470232,        786948,            64;\n  7: 128,    8660012,    346391196,    1660722424,    1660722424,     346391196,       8660012,          128;\n  8: 256,   95268452,   7412784592,   64441336508,  126214904224,   64441336508,    7412784592,     95268452,        256;\n  9: 512, 1047971916, 154448141220, 2283914695884, 7824909176388, 7824909176388, 2283914695884, 154448141220, 1047971916, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(9*n-7)*x)*P(n-1,x) + (9*x-9*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 + (9*n - 7)*x)*P[n - 1, x] + (9*x - 9*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)); 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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2 + (9*n - 7)*x)*v[i - 1] + (9*x - 9*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-6a6e9e081117dc37","steps":[],"edges":[],"text":"A257619"},"headline":"A257619: 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) = 9*x + 2.","chain":null}