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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:                17,                1;\n  2:               452,               54,                1;\n  3:             17348,             3180,              119,              1;\n  4:            908672,           227760,            12992,            220,             1;\n  5:          62312832,         20029376,          1552912,          39920,           365,           1;\n  6:        5418432000,       2142778688,        212024320,        7375872,        102100,         562,         1;\n  7:      582473664000,     275245327872,      33388763712,     1476251840,      27567412,      229292,       819,      1;\n  8:    75828980736000,   41898359098368,    6058231865088,   327172251136,    7740149952,    86651360,    467040,   1144,    1;\n  9: 11752863796224000, 7467107308720128, 1260392761910016, 80877640516864, 2329814281920, 32962698304, 238725632, 881472, 1545, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+11*n^3+45*n^2+81*n+54-(2*n^3+14*n^2+31*n+21)*x+(n^2+2*n-2)*x^2+x^3)*P(n-2,x) - ((2*n^3+17*n^2+47*n+42)*x-(4*n^2+22*n+29)*x^2+(2*n+5)*x^3)*d/dx P(n-2,x) - ((n^2+5*n+6)*x^2-(2*n+5)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+13*n+22-(2*n+6)*x+x^2)*P(n-1,x) + (2*n+9)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 17 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 17 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*D[P[n - 2, x], x] - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*P[n - 1, x] + (2*n + 9)*x*D[P[n - 1, x], x] + x^2*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] = 1; v[2] = 17 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 11*n^3 + 45*n^2 + 81*n + 54 - (2*n^3 + 14*n^2 + 31*n + 21)*x + (n^2 + 2*n - 2)*x^2 + x^3)*v[i - 2] - ((2*n^3 + 17*n^2 + 47*n + 42)*x - (4*n^2 + 22*n + 29)*x^2 + (2*n + 5)*x^3)*deriv(v[i - 2], x) - ((n^2 + 5*n + 6)*x^2 - (2*n + 5)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 13*n + 22 - (2*n + 6)*x + x^2)*v[i - 1] + (2*n + 9)*x*deriv(v[i - 1], x) + x^2*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, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ee73bfb592710e97","steps":["D+1"],"edges":["e-da556ae22956fd27"],"text":"A371081 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