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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"51235c576c0377295d1c83ce57c312d614781ef6595f4c6f1f389fedb8d8e05d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-9c8a3637306e75dc","role":"sequence","name":"Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","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-9c8a3637306e75dc","role":"sequence","name":"Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.","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:   12,      64,        56,          8;\n   2:   24,     248,       468,        204,          16;\n   3:   48,     880,      3060,       2840,         700,          32;\n   4:   96,    2960,     17324,      29084,       15364,        2308,          64;\n   5:  192,    9616,     89332,     246720,      234960,       76928,        7404,         128;\n   6:  384,   30512,    431980,    1844244,     2863640,     1698616,      364668,       23300,         256;\n   7:  768,   95248,   1995188,   12602408,    29871268,    28851872,    11335772,     1661032,       72332,        512;\n   8: 1536,  293936,   8910060,   80611404,   278422908,   408418620,   262173444,    71270820,     7344276,     222372,      1024;\n   9: 3072,  899728,  38801460,  490519760,  2385061640,  5059085696,  4929089984,  2203499600,   428159000,   31751120,    678892,    2048;\n  10: 6144, 2738096, 165780716, 2872142660, 19145934480, 56627959632, 79524214728, 54083931288, 17432954640, 2482785120, 134929324, 2062276, 4096;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(n+1)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (1+(n+3)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 64*x + 56*x^2 + 8*x^3, P(2,x) = 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4.","mathematica":"P[1, x_] := 12 + 64*x + 56*x^2 + 8*x^3;\nP[2, x_] := 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4;\nP[n_, x_] := P[n, x] = Expand[(2 + (n + 1)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (1 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*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] = 12 + 64*x + 56*x^2 + 8*x^3; v[2] = 24 + 248*x + 468*x^2 + 204*x^3 + 16*x^4; for(n = 3, N, my(i = n); v[i] = (2 + (n + 1)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (1 + (n + 3)*x)*v[i - 1] + (x - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9c8a3637306e75dc","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a7f8a3dfd72049c9"],"text":"A256890 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A256890 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}