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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:       1;\n  1:       5,        5,         1;\n  2:      29,       58,        40,        11,          1;\n  3:     169,      507,       601,       357,        111,         17,          1;\n  4:     985,     3940,      6632,      6106,       3346,       1112,        218,         23,         1;\n  5:    5741,    28705,     62633,     78302,      61918,      32258,      11180,       2539,       361,        29,         1;\n  6:   33461,   200766,    538272,    851005,     881763,     629448,     316758,     113001,     28315,      4854,       540,       35,       1;\n  7:  195025,  1365175,   4341905,   8304155,   10653725,    9683875,    6421385,    3150845,   1148035,    308725,     60259,     8273,     755,      41,      1;\n  8: 1136689,  9093512,  33464240,  75113220,  114967140,  127085776,  104827868,   65741290,  31643590,  11716920,   3323376,   713948,  113816,   13012,   1006,    47,    1;\n  9: 6625109, 59625981, 249215921, 642205132, 1142342460, 1488135236, 1470006144, 1124823234, 675266038, 320148710, 120059622, 35503348, 8212400, 1465092, 197086, 19287, 1293, 53, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (16*n^2-42*n+31-(8*n-13)*x+x^2)*P(n-1,x) - (32*n-64+(32*n-78)*x+(4*n-19)*x^2-x^3)*d/dx P(n-1,x) + (16+32*x+18*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 42*n + 31 - (8*n - 13)*x + x^2)*P[n - 1, x] - (32*n - 64 + (32*n - 78)*x + (4*n - 19)*x^2 - x^3)*D[P[n - 1, x], x] + (16 + 32*x + 18*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];\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] = ((16*n^2 - 42*n + 31 - (8*n - 13)*x + x^2)*v[i - 1] - (32*n - 64 + (32*n - 78)*x + (4*n - 19)*x^2 - x^3)*deriv(v[i - 1], x) + (16 + 32*x + 18*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 1)); 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}}],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A008288","x \u21a6 x+1"],"edges":["e-054f97e0823e3ab6","e-f34d07646e032d52"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 x \u21a6 x+1","chain":null}