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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"59fc96a6c16a39bf84f91c2f4109ad011f364e3c3f380443acee24dab406d2b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:             6,             1;\n   2:            50,            16,             1;\n   3:           630,           270,            31,            1;\n   4:         10530,          5430,           855,           51,           1;\n   5:        218790,        128970,         24975,         2075,          76,          1;\n   6:       5429970,       3551310,        799995,        82875,        4275,        106,        1;\n   7:     156633750,     111474090,      28254555,      3443895,      223650,       7875,      141,       1;\n   8:    5148029250,    3933042750,    1097280135,    151911375,    11702145,     522480,    13370,     181,     1;\n   9:  189871431750,  154171467450,   46617053175,   7160282955,   630821520,   33615225,  1096830,   21330,   226,   1;\n  10: 7764742235250, 6649773079950, 2154425548275, 361075309875, 35501320575, 2175063030, 85113315, 2119950, 32400, 276, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-7+x)*P(n-2,x) + x*d/dx P(n-2,x) + (4*n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 6 + x, P(2,x) = 50 + 16*x + x^2.","mathematica":"P[1, x_] := 6 + x;\nP[2, x_] := 50 + 16*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(4*n - 7 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (4*n + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 6 + x; v[2] = 50 + 16*x + x^2; for(n = 3, N, my(i = n); v[i] = (4*n - 7 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (4*n + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5c56442adc0376c2","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-80d7320f7911dda9"],"text":"A049029 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A049029 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}