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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d1cf436749f41d8c3f30272e618eba7b8b6cba8869e8955d66e6005f99fc806b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f2cd67e92f70216b","role":"sequence","name":"Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).","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-f2cd67e92f70216b","role":"sequence","name":"Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).","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:         -1,           1;\n  1:          6,          -5,          1;\n  2:        -40,          38,        -11,           1;\n  3:        336,        -356,        128,         -19,         1;\n  4:      -3456,        3984,      -1660,         320,       -29,         1;\n  5:      42240,      -52064,      24184,       -5540,       670,       -41,       1;\n  6:    -599040,      779904,    -394240,      102424,    -14980,      1246,     -55,       1;\n  7:    9676800,   -13190400,    7139712,    -2044784,    343504,    -35000,    2128,     -71,    1;\n  8: -175472640,   248776704, -142545664,    44162432,  -8259216,    976080,  -73416,    3408,  -89,    1;\n  9: 3530096640, -5179060224, 3114809856, -1030100480, 210065120, -27853392, 2448768, -141720, 5190, -109, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1-x)*P(n,x) = -(4*n^2+2*n-(4*n+1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -1 + x.","mathematica":"P[0, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 + 2*n - (4*n + 1)*x + x^2)*P[n - 1, x])/(2*n - 1 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -1 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^2 + 2*n - (4*n + 1)*x + x^2)*v[i - 1])/(2*n - 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-4f6e747b794bcc2e","steps":["x \u21a6 \u2212x+1","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-8b470702f19ec444","e-769dd7c64d51b4a5"],"text":"A028338 \u2192 x \u21a6 \u2212x+1 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u2212x+1 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
