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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"be8b51ab7ffc66800e95eb27ee27aa51af1acf84320268add8692c81a92603f3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-54aa7a9d2edaadfb","role":"sequence","name":"A triangle of numbers related to triangle A030527.","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-54aa7a9d2edaadfb","role":"sequence","name":"A triangle of numbers related to triangle A030527.","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:           7,            1;\n  1:          48,           19,            1;\n  2:         378,          294,           37,            1;\n  3:        3360,         4476,          996,           61,           1;\n  4:       33264,        70224,        24360,         2520,          91,          1;\n  5:      362880,      1152648,       580824,        89820,        5340,        127,        1;\n  6:     4324320,     19891872,     13907376,      3003504,      260610,      10038,      169,       1;\n  7:    55883520,    361291392,    339204096,     97918128,    11598384,     641928,    17304,     217,     1;\n  8:   778377600,   6901614720,   8488839744,   3175913664,   492007824,   36623664,  1403136,   27936,   271,   1;\n  9: 11623772160, 138468185856, 218796754176, 103671247680, 20384844480, 1956322368, 99747648, 2799720, 42840, 331, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+4+x)*P(n-2,x) + 5*x*d/dx P(n-2,x) + (n+5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 7 + x, P(1,x) = 48 + 19*x + x^2.","mathematica":"P[0, x_] := 7 + x;\nP[1, x_] := 48 + 19*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(n + 4 + x)*P[n - 2, x] + 5*x*D[P[n - 2, x], x] + (n + 5 + x)*P[n - 1, x] + 5*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] = 7 + x; v[2] = 48 + 19*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (n + 4 + x)*v[i - 2] + 5*x*deriv(v[i - 2], x) + (n + 5 + x)*v[i - 1] + 5*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-54aa7a9d2edaadfb","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0a2b33f6b07eac20"],"text":"A049374 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A049374 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}