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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"f75630716d0c026310aaa0197d4c4f7255614f6536f10c1e631a39a44428e53f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-7c9c430277778c10","role":"sequence","name":"Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/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-7c9c430277778c10","role":"sequence","name":"Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/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:         2,          2;\n   2:         4,         10,          4;\n   3:        18,         54,         40,          8;\n   4:       120,        398,        380,        136,         16;\n   5:      1050,       3730,       4144,       1968,        416,        32;\n   6:     11340,      42426,      52356,      29680,       8480,      1184,        64;\n   7:    145530,     567462,     757880,     486872,     168800,     32288,      3200,      128;\n   8:   2162160,    8724894,   12406012,    8763160,    3489584,    819616,    112448,     8320,     256;\n   9:  36486450,  151590690,  226933344,  172867680,   76527808,  20850368,   3544576,   366080,   20992,   512;\n  10: 689188500, 2936803050, 4591521540, 3721744352, 1793105984, 547635648, 108829056, 14028288, 1130496, 51712, 1024;\n","recurrence_sentence":"The row polynomials satisfy (n-1+x)*P(n,x) = (2*n^2-3*n+(4*n-3)*x+2*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2 + 2*x.","mathematica":"P[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 3*n + (4*n - 3)*x + 2*x^2)*P[n - 1, x])/(n - 1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 3*n + (4*n - 3)*x + 2*x^2)*v[i - 1])/(n - 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-7c9c430277778c10","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3d3610943d28dca0"],"text":"A161198 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A161198 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
