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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"a47d805031bc8d8e6fa0da86cd7f22084046ce715307f523cb755ced8d957d05","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-25dfcdb2c8790bf0","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 = 1.","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-25dfcdb2c8790bf0","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 = 1.","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:       -1,         1;\n  1:        4,        -4,        1;\n  2:      -18,        21,       -8,         1;\n  3:       96,      -128,       62,       -13,        1;\n  4:     -600,       890,     -509,       141,      -19,        1;\n  5:     4320,     -6984,     4524,     -1510,      275,      -26,      1;\n  6:   -35280,     61236,   -43756,     16759,    -3730,      484,    -34,      1;\n  7:   322560,   -594432,   460144,   -196252,    50624,    -8113,    791,    -43,    1;\n  8: -3265920,   6336144, -5244732,   2441088,  -706517,   132265, -16058,   1222,  -53,   1;\n  9: 36288000, -73627200, 64537200, -32311376, 10267424, -2170056, 309729, -29544, 1806, -64, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-x)*P(n,x) = -(n^2+2*n+1-(2*n+2)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -1 + x.","mathematica":"P[0, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 2*n + 1 - (2*n + 2)*x + x^2)*P[n - 1, x])/(n - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -1 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 2*n + 1 - (2*n + 2)*x + x^2)*v[i - 1])/(n - 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+3","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-435b07172be58365","e-c7ba3a8cbf32cf64"],"text":"A028338 \u2192 x \u21a6 \u22122x+3 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 \u22122x+3 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}