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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:          12,           9,           1;\n  3:         120,         111,          21,          1;\n  4:        1680,        1785,         447,         38,         1;\n  5:       30240,       35595,       10725,       1245,        60,        1;\n  6:      665280,      848925,      292995,      41850,      2805,       87,      1;\n  7:    17297280,    23586255,     9052785,    1506645,    126000,     5502,    119,     1;\n  8:   518918400,   748471185,   313275375,   58772070,   5790645,   318570,   9786,   156,   1;\n  9: 17643225600, 26715409875, 12026384685, 2487841965, 278816580, 18214875, 710010, 16182, 198, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-2+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[(4*n - 2 + 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] = (4*n - 2 + 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-4172472abe953cea","steps":["basis A008277"],"edges":["e-3697d9eea9f962a2"],"text":"A290319 \u2192 basis A008277"},"headline":"A290319 \u2192 basis A008277","chain":null}