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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:               2;\n  2:              28,               48;\n  3:             662,             2640,              2160;\n  4:           24568,           167328,            302400,             161280;\n  5:         1326122,         13498080,          40007520,           45964800,           18144000;\n  6:        98329108,       1385398608,        5938574400,        10958492160,         9180864000,         2874009600;\n  7:      9596075582,     177932729520,     1027378432080,      2693417126400,      3577942368000,      2353813862400,       610248038400;\n  8:   1192744081648,   28073116365888,   208457658864000,    724868256752640,   1352654214912000,   1398688504012800,    756707567616000,    167382319104000;\n  9: 183983154281042, 5348776806240960, 49426658023515840, 218061839231308800, 534339995821632000, 770096309946163200, 650712365330227200, 298777439600640000, 57621363351552000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+24*x+24*x^2)*P(n-2,x) - (6+32*x+62*x^2+36*x^3)*d/dx P(n-2,x) - (4*x+16*x^2+20*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (14+24*x)*P(n-1,x) + (6+28*x+22*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(9 + 24*x + 24*x^2)*P[n - 2, x] - (6 + 32*x + 62*x^2 + 36*x^3)*D[P[n - 2, x], x] - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (14 + 24*x)*P[n - 1, x] + (6 + 28*x + 22*x^2)*D[P[n - 1, x], x] + (4*x + 8*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 24*x + 24*x^2)*v[i - 2] - (6 + 32*x + 62*x^2 + 36*x^3)*deriv(v[i - 2], x) - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (14 + 24*x)*v[i - 1] + (6 + 28*x + 22*x^2)*deriv(v[i - 1], x) + (4*x + 8*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, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3338feb205e3b77f","steps":["D"],"edges":["e-4059c9cd4da3a5f0"],"text":"A151775 \u2192 D"},"headline":"A151775 \u2192 D","chain":null}