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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"76bf44849787420b0f58df286e5840e746b1fd54237465d82d7c3f195957c907","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-95a19db879d4e5b4","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 = 5.","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-95a19db879d4e5b4","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 = 5.","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:          -5,            1;\n  1:          36,          -12,          1;\n  2:        -294,          133,        -20,           1;\n  3:        2688,        -1504,        314,         -29,         1;\n  4:      -27216,        17874,      -4665,         605,       -39,         1;\n  5:      302400,      -225480,      69524,      -11350,      1035,       -50,       1;\n  6:    -3659040,      3027684,   -1064768,      206359,    -23810,      1636,     -62,       1;\n  7:    47900160,    -43263936,   16944192,    -3759420,    516964,    -45129,    2443,     -75,    1;\n  8:  -674593920,    656867952, -281621916,    69794416, -11021801,   1150457,  -79394,    3494,  -89,    1;\n  9: 10170800640, -10574149824, 4899280176, -1332515920, 235663280, -28325472, 2343873, -131880, 4830, -104, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+4-x)*P(n,x) = -(n^2+10*n+25-(2*n+10)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -5 + x.","mathematica":"P[0, x_] := -5 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 10*n + 25 - (2*n + 10)*x + x^2)*P[n - 1, x])/(n + 4 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -5 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 10*n + 25 - (2*n + 10)*x + x^2)*v[i - 1])/(n + 4 - 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+11","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2e6c2042efefd194","e-8a24318fab51e026"],"text":"A028338 \u2192 x \u21a6 \u22122x+11 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+11 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}