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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":"08ffddd80a0225e99b36d566d89165176167ac2e4757fa1bdfc2bc8074628109","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cfa0bcf0bbd4ed6d","role":"sequence","name":"Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) terms per row.","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;\n  2: 1,   2;\n  3: 1,   6;\n  4: 1,  14,    3;\n  5: 1,  30,   18;\n  6: 1,  62,   75,    4;\n  7: 1, 126,  270,   40;\n  8: 1, 254,  903,  260,  5;\n  9: 1, 510, 2898, 1400, 75;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - x^2*d/dx P(n-3,x) + 2*x*P(n-2,x) + P(n-1,x) + x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 1 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - x^2*D[P[n - 3, x], x] + 2*x*P[n - 2, x] + P[n - 1, x] + x*D[P[n - 1, x], 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; v[3] = 1 + 2*x; v[4] = 1 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - x^2*deriv(v[i - 3], x) + 2*x*v[i - 2] + v[i - 1] + x*deriv(v[i - 1], 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-cfa0bcf0bbd4ed6d","steps":["\u03b8+1"],"edges":["e-5eb8bb2bfa60e69d"],"text":"A136011 \u2192 \u03b8+1"},"headline":"A136011 \u2192 \u03b8+1","chain":null}