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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:        2,           1;\n  2:       13,          14,            1;\n  3:      104,         241,           72,             1;\n  4:      835,        5174,         3118,           320,            1;\n  5:     6686,      114505,       137400,         33146,         1334,            1;\n  6:    53497,     2491526,      6175915,       2777080,       308255,         5418,          1;\n  7:   427988,    53338489,    272272836,     221141895,     47240260,      2618603,      21788,         1;\n  8:  3423919,  1130807966,  11666247472,   17009865456,   6261487320,    708720534,   20923392,     87308,      1;\n  9: 27391370, 23853108769, 488145521988, 1260684623920, 773443027716, 151011601338, 9672858564, 160346244, 349434, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-4+4*x)*P(n,x) = -(8*n-24+(6*n^2-47*n+119)*x+(9*n^3-96*n^2+327*n-404)*x^2+(54*n^2-360*n+474)*x^3+(72*n-120)*x^4)*P(n-2,x) - ((13*n-39)*x+(30*n^2-213*n+421)*x^2-(9*n^2-225*n+726)*x^3-(54*n-366)*x^4-72*x^5)*d/dx P(n-2,x) - ((3*n-9)*x^2+(9*n^2-63*n+120)*x^3+(54*n-198)*x^4+72*x^5)*d^2/dx^2 P(n-2,x) + (9*n-35+(6*n^2-47*n+128)*x+(30*n-122)*x^2+24*x^3)*P(n-1,x) + ((13*n-52)*x-(3*n-64)*x^2-12*x^3)*d/dx P(n-1,x) + ((3*n-12)*x^2+12*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 24 + (6*n^2 - 47*n + 119)*x + (9*n^3 - 96*n^2 + 327*n - 404)*x^2 + (54*n^2 - 360*n + 474)*x^3 + (72*n - 120)*x^4)*P[n - 2, x] - ((13*n - 39)*x + (30*n^2 - 213*n + 421)*x^2 - (9*n^2 - 225*n + 726)*x^3 - (54*n - 366)*x^4 - 72*x^5)*D[P[n - 2, x], x] - ((3*n - 9)*x^2 + (9*n^2 - 63*n + 120)*x^3 + (54*n - 198)*x^4 + 72*x^5)*D[P[n - 2, x], {x, 2}] + (9*n - 35 + (6*n^2 - 47*n + 128)*x + (30*n - 122)*x^2 + 24*x^3)*P[n - 1, x] + ((13*n - 52)*x - (3*n - 64)*x^2 - 12*x^3)*D[P[n - 1, x], x] + ((3*n - 12)*x^2 + 12*x^3)*D[P[n - 1, x], {x, 2}])/(n - 4 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n - 24 + (6*n^2 - 47*n + 119)*x + (9*n^3 - 96*n^2 + 327*n - 404)*x^2 + (54*n^2 - 360*n + 474)*x^3 + (72*n - 120)*x^4)*v[i - 2] - ((13*n - 39)*x + (30*n^2 - 213*n + 421)*x^2 - (9*n^2 - 225*n + 726)*x^3 - (54*n - 366)*x^4 - 72*x^5)*deriv(v[i - 2], x) - ((3*n - 9)*x^2 + (9*n^2 - 63*n + 120)*x^3 + (54*n - 198)*x^4 + 72*x^5)*deriv(deriv(v[i - 2], x), x) + (9*n - 35 + (6*n^2 - 47*n + 128)*x + (30*n - 122)*x^2 + 24*x^3)*v[i - 1] + ((13*n - 52)*x - (3*n - 64)*x^2 - 12*x^3)*deriv(v[i - 1], x) + ((3*n - 12)*x^2 + 12*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 4 + 4*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-25ac1cc25c118491","steps":["D+1"],"edges":["e-5413d412f5e0676c"],"text":"A166962 \u2192 D+1"},"headline":"A166962 \u2192 D+1","chain":null}