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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"a1b49105c59b3afe1cf0d25eaf1858101146f1ae524bd612f2b3b1573878dcc6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-64714e0ad28cd8ed","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(x) = 7*x + 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-64714e0ad28cd8ed","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(x) = 7*x + 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,         63,             9;\n  2:    36,        813,           762,             27;\n  3:   108,       8931,         26355,           8205,              81;\n  4:   324,      92469,        657108,         639684,           84372,             243;\n  5:   972,     936435,      13980129,       30943794,        13432914,          852387,             729;\n  6:  2916,    9406389,     272656686,     1171614513,      1154650848,       260160207,         8554974,           2187;\n  7:  8748,   94210419,    5052486159,    38386262019,     71902040175,     37504267857,      4801732341,       85658361,          6561;\n  8: 26244,  942605013,   90731326776,  1146817009170,   3679652086572,   3646136308416,   1114365227472,    86023804854,     856955400,      19683;\n  9: 78732, 9427736307, 1597446657741, 32208084965760, 165415265377824, 277884633843894, 162915946078578, 31184451065064, 1512365457900, 8570807151, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(7*n-11)*x)*P(n-2,x) + (7*x-7*x^2)*d/dx P(n-2,x) + (2+(7*n+3)*x)*P(n-1,x) + (7*x-7*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 4 + 3*x, P(1,x) = 12 + 63*x + 9*x^2.","mathematica":"P[0, x_] := 4 + 3*x;\nP[1, x_] := 12 + 63*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (7*n - 11)*x)*P[n - 2, x] + (7*x - 7*x^2)*D[P[n - 2, x], x] + (2 + (7*n + 3)*x)*P[n - 1, x] + (7*x - 7*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 + 63*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = (3 + (7*n - 11)*x)*v[i - 2] + (7*x - 7*x^2)*deriv(v[i - 2], x) + (2 + (7*n + 3)*x)*v[i - 1] + (7*x - 7*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-64714e0ad28cd8ed","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-68ca759bf35f8a33"],"text":"A257627 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257627 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}