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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,           8;\n  2:         10,          56,          48;\n  3:         80,         528,         720,         256;\n  4:        880,        6448,       11088,        6656,        1280;\n  5:      12320,       97312,      193920,      152320,       51200,        6144;\n  6:     209440,     1752864,     3880512,     3623680,     1632000,      350208,      28672;\n  7:    4188800,    36732800,    88127424,    93169664,    50758400,    14837760,    2207744,    131072;\n  8:   96342400,   878364800,  2247327552,  2612915200,  1633291520,   584908800,  120020992,  13107200,   589824;\n  9: 2504902400, 23608224000, 63700705152, 79921542144, 55530155520, 23047428096, 5850120192, 889454592, 74317824, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(24*n-32)*x+16*x^2)*P(n-2,x) + (6*n-5+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*P[n - 2, x] + (6*n - 5 + 8*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (24*n - 32)*x + 16*x^2)*v[i - 2] + (6*n - 5 + 8*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-260e022530e5976c","steps":["basis A013611","\u03b8+1"],"edges":["e-10584eb473514fc3","e-272c682965e45e4c"],"text":"A286718 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A286718 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}