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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"7e55a73b75661609b6baf17591491ab3e84d5535ab04d573a26aa0f25f3b0c47","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0778fc04c5956f01","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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;\n  1:       5,        6;\n  2:      24,       60,       27;\n  3:     115,      438,      405,       108;\n  4:     551,     2820,     3969,      2160,       405;\n  5:    2640,    16968,    32130,     26568,     10125,      1458;\n  6:   12649,    97860,   233037,    259200,    149850,     43740,     5103;\n  7:   60605,   548226,  1573425,   2201580,   1711125,    756702,   178605,    17496;\n  8:  290376,  3006900, 10101105,  17042400,  16661700,   9899820,  3536379,   699840,   59049;\n  9: 1391275, 16228530, 62463150, 123414840, 145506375, 108724518, 52152660, 15606432, 2657205, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(28*n+10*n*x-6*n^3*x^2)*P(n,x) = -(70*n+70+(21*n^2+109*n+88)*x-(15*n^3+15*n^2-30*n-30)*x^2-(9*n^3+18*n^2+9*n)*x^3)*P(n-1,x) - (49+105*x-(21*n^2+21*n-71)*x^2-(30*n^2+30*n-15)*x^3-(9*n^2+9*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 70 + (21*n^2 + 109*n + 88)*x - (15*n^3 + 15*n^2 - 30*n - 30)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*P[n - 1, x] - (49 + 105*x - (21*n^2 + 21*n - 71)*x^2 - (30*n^2 + 30*n - 15)*x^3 - (9*n^2 + 9*n)*x^4)*D[P[n - 1, x], x])/(-28*n - 10*n*x + 6*n^3*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(70*n + 70 + (21*n^2 + 109*n + 88)*x - (15*n^3 + 15*n^2 - 30*n - 30)*x^2 - (9*n^3 + 18*n^2 + 9*n)*x^3)*v[i - 1] - (49 + 105*x - (21*n^2 + 21*n - 71)*x^2 - (30*n^2 + 30*n - 15)*x^3 - (9*n^2 + 9*n)*x^4)*deriv(v[i - 1], x))/(-28*n - 10*n*x + 6*n^3*x^2)); 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-30cb3cad216ed189","steps":["basis A013610","\u03b8+1"],"edges":["e-99fce6e1b50f281f","e-83352fbac5da0e3c"],"text":"A207823 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A207823 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}