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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (9*n - 8)*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};\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":"A257608","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: 1,         1;\n  2: 1,        20,           1;\n  3: 1,       219,         219,            1;\n  4: 1,      2218,        8322,         2218,             1;\n  5: 1,     22217,      220222,       220222,         22217,             1;\n  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;\n  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;\n  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;\n  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(9*n-8)*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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*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] = (1 + (9*n - 8)*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-dcd2d11144e6ea21","steps":[],"edges":[],"text":"A257608"},"headline":"A257608: 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 + 1.","chain":null}