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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:                  3,                   2;\n  2:                 57,                  76,                  24;\n  3:               2763,                5462,                3480,                 720;\n  4:             250737,              655576,              627984,              262080,               40320;\n  5:           36581523,           118940522,           151991760,            95588640,            29635200,             3628800;\n  6:         7828053417,         30435667876,         48615763944,         40879776960,         19105511040,          4710182400,           479001600;\n  7:      2309644635483,      10450328632382,      20027633993640,      21089047739760,      13187071497600,       4900146451200,       1002550348800,        87178291200;\n  8:    898621108880097,    4638024381708976,   10367787111743904,   13117171658048640,   10278725669372160,    5111018283571200,    1575660435148800,    275483400192000,    20922789888000;\n  9: 445777636063460643, 2584568731265561042, 6601458159781401120, 9753008378506390080, 9188659740573331200, 5727242569227379200, 2362582180204646400, 622221802339737600, 94968543301632000, 6402373705728000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + (24+32*x+10*x^2)*d/dx P(n-1,x) + (16+32*x+20*x^2+4*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[(3 + 2*x)*P[n - 1, x] + (24 + 32*x + 10*x^2)*D[P[n - 1, x], x] + (16 + 32*x + 20*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] + (24 + 32*x + 10*x^2)*deriv(v[i - 1], x) + (16 + 32*x + 20*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":["x \u21a6 x+1"],"edges":["e-97a8f91248a4ec37"],"text":"A151775 \u2192 x \u21a6 x+1"},"headline":"A151775 \u2192 x \u21a6 x+1","chain":null}