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sequences.","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["4","4"]],"card_rows":[["1"],["4","4"],["9","27","9"],["16","112","96","16"]],"row_sums_preview":["0","1","8","45","240","1300","7308","42973"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1c11ffc3c4cd47ed","steps":["\u03b8+1","basis A008277"],"edges":["e-0695c41826b28137","e-cd93b1c302bdfd1b"],"text":"A132440 \u2192 \u03b8+1 \u2192 basis A008277"},"headline":"A132440 \u2192 \u03b8+1 \u2192 basis A008277","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   9,     27,        9;\n   2:  16,    112,       96,       16;\n   3:  25,    375,      625,      250,        25;\n   4:  36,   1116,     3240,     2340,       540,        36;\n   5:  49,   3087,    14749,    17150,      6860,      1029,       49;\n   6:  64,   8128,    61824,   108864,     67200,     17024,     1792,       64;\n   7:  81,  20655,   245025,   629370,    563031,    214326,    37422,     2916,      81;\n   8: 100,  51100,   933000,  3410500,   4252500,   2282700,   588000,    75000,    4500,    100;\n   9: 121, 123783,  3448621, 17635750,  29854330,  21717927,  7742427,  1437480,  139755,   6655,  121;\n  10: 144, 294768, 12459744, 88056144, 198633600, 190605888, 90345024, 22899888, 3207600, 245520, 9504, 144;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (n^2+4*n+4+(n^2+4*n+4)*x)*P(n-1,x) + (n^2+4*n+4)*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 27*x + 9*x^2.","mathematica":"P[1, x_] := 9 + 27*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 4 + (n^2 + 4*n + 4)*x)*P[n - 1, x] + (n^2 + 4*n + 4)*x*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 9 + 27*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = ((n^2 + 4*n + 4 + (n^2 + 4*n + 4)*x)*v[i - 1] + (n^2 + 4*n + 4)*x*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5fe240122e98ffb3","steps":["(\u03b8+1)\u00b2","basis A008277"],"edges":["e-c527a11d264443d6","e-41d9df5353042853"],"text":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277"},"headline":"basis-map coefficients of A112227 \u2192 (\u03b8+1)\u00b2 \u2192 basis A008277","chain":null}