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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"81a4b6f887bbaa8e8c65ac97e6c57e7b722f44b25948aa7b56412d12ad70abfc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","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-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","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:          4872,           1155,             64,              1;\n  1:         58968,          24297,           2394,             85,             1;\n  2:        653184,         435456,          67536,           4074,           106,            1;\n  3:       6757344,        6944112,        1583253,         143850,          6195,          127,           1;\n  4:      66204864,      101150208,       32430888,        4150629,        262500,         8757,         148,          1;\n  5:     620478144,     1369540224,      598274856,      103312881,       8980335,       432747,       11760,        169,        1;\n  6:    5604318720,    17451023616,    10141103232,     2292428736,     262514826,     17109603,      663852,      15204,      190,       1;\n  7:   49063262976,   211206393216,   160202543904,    46347017184,    6787922661,    573770925,    29770146,     965076,    19089,     211,     1;\n  8:  418184073216,  2445215205888,  2383504014528,   867056576256,  158881287240,  16915659111,  1125708948,   48388158,  1345680,   23415,   232,   1;\n  9: 3482609872896, 27232873920000, 33668882356032, 15182785836288, 3422039202360, 449077969881, 37235476656, 2035934334, 74584314, 1814925, 28182, 253, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+4)*P(n,x) = (6*n+42+2*x)*P(n-2,x) + (36*x+x^2)*d/dx P(n-2,x) + (5*n+52+2*x)*P(n-1,x) + (36*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4872 + 1155*x + 64*x^2 + x^3, P(1,x) = 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4.","mathematica":"P[0, x_] := 4872 + 1155*x + 64*x^2 + x^3;\nP[1, x_] := 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 42 + 2*x)*P[n - 2, x] + (36*x + x^2)*D[P[n - 2, x], x] + (5*n + 52 + 2*x)*P[n - 1, x] + (36*x + x^2)*D[P[n - 1, x], x])/(n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4872 + 1155*x + 64*x^2 + x^3; v[2] = 58968 + 24297*x + 2394*x^2 + 85*x^3 + x^4; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 42 + 2*x)*v[i - 2] + (36*x + x^2)*deriv(v[i - 2], x) + (5*n + 52 + 2*x)*v[i - 1] + (36*x + x^2)*deriv(v[i - 1], x))/(n + 4)); 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-44b1dbf5d009bf35","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0a71e6005af9c857"],"text":"A132166 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A132166 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}