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0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 0;\n  1: 1;\n  2: 1,     24;\n  3: 1,    120,       1080;\n  4: 1,    504,      15120,        80640;\n  5: 1,   2040,     158760,      2419200,        9072000;\n  6: 1,   8184,    1520640,     50561280,      498960000,      1437004800;\n  7: 1,  32760,   14054040,    922521600,    18162144000,    130767436800,     305124019200;\n  8: 1, 131064,  127960560,  15809713920,   557837280000,   7584511334400,   42717362688000,    83691159552000;\n  9: 1, 524280, 1157542920, 262509811200, 15724524816000, 361403833190400, 3703595345049600, 17072996548608000, 28810681675776000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -24*x^2*P(n-2,x) - (4*x^2+36*x^3)*d/dx P(n-2,x) - (2*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (1+24*x)*P(n-1,x) + (3*x+22*x^2)*d/dx P(n-1,x) + (x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-24*x^2*P[n - 2, x] - (4*x^2 + 36*x^3)*D[P[n - 2, x], x] - (2*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 24*x)*P[n - 1, x] + (3*x + 22*x^2)*D[P[n - 1, x], x] + (x^2 + 4*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -24*x^2*v[i - 2] - (4*x^2 + 36*x^3)*deriv(v[i - 2], x) - (2*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 24*x)*v[i - 1] + (3*x + 22*x^2)*deriv(v[i - 1], x) + (x^2 + 4*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, 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