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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":"9e91fc0b9ea21fa3fb41d5939b50223e458322eec1179502f8189810afe51ca1","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-dabecbb1addaf41b","role":"sequence","name":"basis_map_A013610 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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:      1,       6;\n  2:      2,      18,       27;\n  3:      6,      66,      162,      108;\n  4:     24,     300,      945,     1080,       405;\n  5:    120,    1644,     6075,     9180,      6075,     1458;\n  6:    720,   10584,    43848,    79380,     70875,    30618,     5103;\n  7:   5040,   78408,   354564,   731052,    793800,   469476,   142884,    17496;\n  8:  40320,  657504,  3189348,  7266672,   9091845,  6613488,  2786238,   629856,   59049;\n  9: 362880, 6159456, 31662900, 78157440, 109076625, 92252034, 48223350, 15221520, 2657205, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(6*n-6)*x+9*x^2)*P(n-2,x) + (2*n-1+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (6*n - 6)*x + 9*x^2)*P[n - 2, x] + (2*n - 1 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (6*n - 6)*x + 9*x^2)*v[i - 2] + (2*n - 1 + 6*x)*v[i - 1]); 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-b9fbfb57d9269e2f","steps":["basis A013610","\u03b8+1"],"edges":["e-3e76251141af3684","e-af60f7860d45f10c"],"text":"A132393 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A132393 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}