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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"89fd347a5f18428ddfb4c9b5d809e236babe367b076cf7ccf2bf4c8661ef1cb6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f055ce55e9689b65","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 8.","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-f055ce55e9689b65","role":"sequence","name":"Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 8.","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:           -8,             1;\n  1:           81,           -18,           1;\n  2:         -900,           280,         -29,           1;\n  3:        10890,         -4279,         629,         -41,          1;\n  4:      -142560,         66816,      -12486,        1163,        -54,          1;\n  5:      2007720,      -1082562,      242279,      -28810,       1920,        -68,       1;\n  6:    -30270240,      18317544,    -4729892,      675634,     -57665,       2941,     -83,       1;\n  7:    486486000,    -324504900,    94251240,   -15570051,    1600249,    -104790,    4270,     -99,    1;\n  8:  -8302694400,    6022540800, -1931761296,   359647756,  -42833924,    3384689, -177464,    5954, -116,    1;\n  9: 149967417600, -117052405200, 40893480684, -8421696604, 1132318277, -103866126, 6583395, -284736, 8043, -134, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+7-x)*P(n,x) = -(n^2+16*n+64-(2*n+16)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -8 + x.","mathematica":"P[0, x_] := -8 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 16*n + 64 - (2*n + 16)*x + x^2)*P[n - 1, x])/(n + 7 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -8 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 16*n + 64 - (2*n + 16)*x + x^2)*v[i - 1])/(n + 7 - 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 \u22122x+17","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-86838a696acab184","e-319662bfd9739006"],"text":"A028338 \u2192 x \u21a6 \u22122x+17 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+17 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}