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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"c8694901a5e1ee741bc89ad0a4b8be8ee6c3815dcf5790116fa2864f6678eab3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-880485a065802fcb","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 = 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-880485a065802fcb","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 = 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:        -2,          1;\n  1:         9,         -6,         1;\n  2:       -48,         40,       -11,          1;\n  3:       300,       -295,       107,        -17,        1;\n  4:     -2160,       2412,     -1056,        227,      -24,        1;\n  5:     17640,     -21798,     10979,      -2890,      420,      -32,      1;\n  6:   -161280,     216576,   -121784,      37234,    -6695,      709,    -41,      1;\n  7:   1632960,   -2351592,   1446570,    -497259,   104629,   -13818,   1120,    -51,    1;\n  8: -18144000,   27741600, -18397800,    6956788, -1655318,   257369, -26180,   1682,  -62,   1;\n  9: 219542400, -353635920, 250095708, -102416884, 26932373, -4758306, 572691, -46416, 2427, -74, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+1-x)*P(n,x) = -(n^2+4*n+4-(2*n+4)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -2 + x.","mathematica":"P[0, x_] := -2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 4*n + 4 - (2*n + 4)*x + x^2)*P[n - 1, x])/(n + 1 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -2 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 4*n + 4 - (2*n + 4)*x + 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 - 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+5","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-88d2cf9ed2610e27","e-6ddcf5c6f38774e7"],"text":"A028338 \u2192 x \u21a6 \u22122x+5 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+5 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
