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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":"0b468384aa4ad9efb20f406b6feb98f832b394b0a2a2b5a70ebbee4a967a3395","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-551f44a89cf7b3e1","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+5) 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:       6,        6;\n  2:      35,       72,        27;\n  3:     204,      636,       486,       108;\n  4:    1189,     4968,      5751,      2592,       405;\n  5:    6930,    36306,     56376,     38448,     12150,      1458;\n  6:   40391,   254448,    495882,    453600,    216675,     52488,     5103;\n  7:  235416,  1732728,   4063932,   4666680,   2988900,   1093500,   214326,    17496;\n  8: 1372105, 11554416,  31684986,  43802208,  35216775,  17268552,  5108103,   839808,   59049;\n  9: 7997214, 75826398, 238040856, 384886512, 372570030, 229298202, 90874224, 22534848, 3188646, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(384*n+108*n*x-54*n^3*x^2)*P(n,x) = -(1152*n+1152+(288*n^2+1476*n+1188)*x-(162*n^3+162*n^2-324*n-324)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (1024+1728*x-(288*n^2+288*n-936)*x^2-(324*n^2+324*n-162)*x^3-(81*n^2+81*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[(-(1152*n + 1152 + (288*n^2 + 1476*n + 1188)*x - (162*n^3 + 162*n^2 - 324*n - 324)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (1024 + 1728*x - (288*n^2 + 288*n - 936)*x^2 - (324*n^2 + 324*n - 162)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-384*n - 108*n*x + 54*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] = (-(1152*n + 1152 + (288*n^2 + 1476*n + 1188)*x - (162*n^3 + 162*n^2 - 324*n - 324)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (1024 + 1728*x - (288*n^2 + 288*n - 936)*x^2 - (324*n^2 + 324*n - 162)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-384*n - 108*n*x + 54*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-c2b7c74f8491f01b","steps":["basis A013610","\u03b8+1"],"edges":["e-d13df69ca10128cd","e-2de0c38cff2f5d67"],"text":"A207824 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A207824 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}