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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"ee5b93ee78c27546f2d0feef66d0636a0dfa4dd85e1d72148d76af49de64bdb7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-238e6ef5447963d1","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 = 4.","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-238e6ef5447963d1","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 = 4.","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:         -4,           1;\n  1:         25,         -10,          1;\n  2:       -180,          96,        -17,          1;\n  3:       1470,        -959,        233,        -25,         1;\n  4:     -13440,       10208,      -3074,        459,       -34,         1;\n  5:     136080,     -116586,      41199,      -7690,       800,       -44,       1;\n  6:   -1512000,     1429800,    -573100,     126274,    -16525,      1285,     -55,      1;\n  7:   18295200,   -18797460,    8351524,   -2096563,    325409,    -31990,    1946,    -67,    1;\n  8: -239500800,   264219840, -127984896,   35741292,  -6344240,    742609,  -57344,   2818,  -80,   1;\n  9: 3372969600, -3958933680, 2064977532, -630593996, 124903421, -16774086, 1547427, -96864, 3939, -94, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+3-x)*P(n,x) = -(n^2+8*n+16-(2*n+8)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -4 + x.","mathematica":"P[0, x_] := -4 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 8*n + 16 - (2*n + 8)*x + x^2)*P[n - 1, x])/(n + 3 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -4 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 8*n + 16 - (2*n + 8)*x + x^2)*v[i - 1])/(n + 3 - 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+9","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-6b900047034f4def","e-6d01d329dd41619f"],"text":"A028338 \u2192 x \u21a6 \u22122x+9 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+9 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}