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A generalized Stirling1 triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 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:         10,          8,          1;\n  3:         80,         82,         18,         1;\n  4:        880,       1064,        316,        32,        1;\n  5:      12320,      16840,       6120,       860,       50,       1;\n  6:     209440,     315440,     133120,     23320,     1910,      72,      1;\n  7:    4188800,    6833680,    3244080,    669480,    69160,    3710,     98,     1;\n  8:   96342400,  168197120,   87935680,  20650560,  2536800,  173040,   6552,   128,   1;\n  9: 2504902400, 4637664640, 2630396160, 686801920, 96754560, 7901040, 382704, 10776, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-1+x)*P(n-1,x) + x*d/dx 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*n - 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\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*n - 1 + x)*v[i - 1] + x*deriv(v[i - 1], 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-260e022530e5976c","steps":["basis A008277"],"edges":["e-74e855bf9275ded5"],"text":"A286718 \u2192 basis A008277"},"headline":"A286718 \u2192 basis A008277","chain":null}