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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:               1,                1;\n  2:               6,               12,                3;\n  3:             120,              300,              135,              15;\n  4:            5040,            14280,             8400,            1680,             105;\n  5:          362880,          1118880,           774900,          207900,           23625,            945;\n  6:        39916800,        131060160,        101704680,        32848200,         5093550,         374220,         10395;\n  7:      6227020800,      21483221760,      18162144000,      6689723040,      1267566300,      128648520,       6621615,      135135;\n  8:   1307674368000,    4698287193600,    4250720073600,   1729944216000,    378453675600,    47448601200,    3405402000,   129729600,    2027025;\n  9: 355687428096000, 1322395045171200, 1264886951328000, 557287745414400, 136080545000400, 19880055795600, 1777279303800, 95108013000, 2791213425, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(16*n^3-64*n^2+76*n-24)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-16*n+6+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*v[i - 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-1336d058dca9979e","steps":["D+1"],"edges":["e-78a0ffcfaa54334f"],"text":"A204420 \u2192 D+1"},"headline":"A204420 \u2192 D+1","chain":null}