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A generalized Stirling1 triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-260e022530e5976c","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:           2,           1;\n  1:           5,           6,           1;\n  2:          32,          44,          13,          1;\n  3:         308,         457,         171,         23,         1;\n  4:        3920,        6132,        2644,        467,        36,         1;\n  5:       61880,      100778,       47959,      10050,      1040,        52,       1;\n  6:     1164800,     1959520,     1004544,     237574,     29705,      2025,      71,      1;\n  7:    25450880,    43989048,    23923066,    6202357,    888069,     74102,    3584,     93,    1;\n  8:   632952320,  1119583712,   639188488,  178298084,  28317982,   2734697,  163492,   5906,  118,   1;\n  9: 17649632000, 31854922480, 18947969388, 5613611676, 968623473, 104673114, 7301931, 328584, 9207, 146, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n-1+x)*P(n,x) = (9*n^2-4+6*n*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 4 + 6*n*x + x^2)*P[n - 1, x])/(3*n - 1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 - 4 + 6*n*x + x^2)*v[i - 1])/(3*n - 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":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-38cf740a66d8523a"],"text":"A286718 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A286718 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
