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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"f62add4127ae437f0e9c05b25ffebf66e89e213ac3009e0d135d6cfcc8870f0f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4155b181fd90d601","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.","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-4155b181fd90d601","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.","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:     4,          3;\n  1:    12,         51,            9;\n  2:    36,        549,          510,            27;\n  3:   108,       4995,        13527,          4521,             81;\n  4:   324,      42309,       263256,        254724,          37896,            243;\n  5:   972,     347139,      4372965,       9258510,        4160550,         309567,            729;\n  6:  2916,    2807973,     66379362,     265980225,      261094680,       62565951,        2499378,          2187;\n  7:  8748,   22570947,    954048195,    6625652511,    12077681415,     6430937193,      894290217,      20074485,         6561;\n  8: 26244,  180932805,  13247829276,  150434052498,   462020083128,   456568054224,   144957163500,   12376110006,    160867068,      19683;\n  9: 78732, 1448689347, 179901529713, 3206771499612, 15558942241548, 25661364298506, 15252808239822, 3075062165388, 167706352512, 1287848523, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(5*n-7)*x)*P(n-2,x) + (5*x-5*x^2)*d/dx P(n-2,x) + (2+(5*n+3)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 51*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 51*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (5*n - 7)*x)*P[n - 2, x] + (5*x - 5*x^2)*D[P[n - 2, x], x] + (2 + (5*n + 3)*x)*P[n - 1, x] + (5*x - 5*x^2)*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] = 4 + 3*x; v[2] = 12 + 51*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (5*n - 7)*x)*v[i - 2] + (5*x - 5*x^2)*deriv(v[i - 2], x) + (2 + (5*n + 3)*x)*v[i - 1] + (5*x - 5*x^2)*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-4155b181fd90d601","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-fccdbe2b7271fb46"],"text":"A257623 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257623 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}