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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"a482da85ac9a167efb983fee27d406ab7fcce8d53cf0d9f8e21822bd54ac9018","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-1b5bd96740815d9f","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 = 6.","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-1b5bd96740815d9f","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 = 6.","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:          -6,            1;\n  1:          49,          -14,           1;\n  2:        -448,          176,         -23,           1;\n  3:        4536,        -2223,         407,         -33,         1;\n  4:      -50400,        29180,       -6724,         771,       -44,         1;\n  5:      609840,      -402974,      110299,      -16010,      1300,       -56,       1;\n  6:    -7983360,      5880096,    -1844016,      319234,    -32955,      2029,     -69,       1;\n  7:   112432320,    -90739272,    31813774,    -6330107,    781949,    -61418,    2996,     -83,    1;\n  8: -1695133440,   1479836064,  -569907352,   127101428, -18093642,   1705305, -106428,    4242,  -98,    1;\n  9: 27243216000, -25469564400, 10632128940, -2610196356, 417415949, -45442026, 3411219, -174384, 5811, -114, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+5-x)*P(n,x) = -(n^2+12*n+36-(2*n+12)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -6 + x.","mathematica":"P[0, x_] := -6 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 12*n + 36 - (2*n + 12)*x + x^2)*P[n - 1, x])/(n + 5 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -6 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 12*n + 36 - (2*n + 12)*x + x^2)*v[i - 1])/(n + 5 - 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+13","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-297f51c747adaedd","e-e24e4dcabf4aa882"],"text":"A028338 \u2192 x \u21a6 \u22122x+13 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+13 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}