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A013609","x \u21a6 x/(x+1)"],"edges":["e-95924f9cf72f1e84","e-1f8e28813e5553dc"],"text":"A013609 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A013609 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:             9;\n  2:           378,            162;\n  3:         11907,          10206,           2187;\n  4:        333396,         428652,         183708,          26244;\n  5:       8751645,       15002820,        9644670,        2755620,        295245;\n  6:     220541454,      472588830,      405076140,      173604060,      37200870,       3188646;\n  7:    5403265623,    13894111602,    14886548145,     8506598940,    2734263945,     468730962,     33480783;\n  8:  129678374952,   389035124856,   500188017672,   357277155480,  153118780920,   39373400808,   5624771544,   344373768;\n  9: 3063651608241, 10503948371112, 15755922556668, 13505076477144, 7234862398470, 2480524250904, 531540910908, 65086642152, 3486784401;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (21*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[Together[((21*n + 9*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 9*n*x)*v[i - 1])/(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-f485cd65a8f8089f","steps":["x \u21a6 x+1","D"],"edges":["e-62ade03cd9f25f04","e-5fe52ba8cb7bc1c7"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 D","chain":null}