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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":"88287f296fbfaa4ad508094261f9ee40998a242d12ae2b3724846e3e87c9ed13","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,    4;\n  3: 1,   12;\n  4: 1,   28,    9;\n  5: 1,   60,   54;\n  6: 1,  124,  225,   16;\n  7: 1,  252,  810,  160;\n  8: 1,  508, 2709, 1040,  25;\n  9: 1, 1020, 8694, 5600, 375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - 4*x*P(n-3,x) + 4*x^2*d/dx P(n-3,x) + 4*x^3*d^2/dx^2 P(n-3,x) + 4*x*P(n-2,x) - (2*x+4*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - 4*x*P[n - 3, x] + 4*x^2*D[P[n - 3, x], x] + 4*x^3*D[P[n - 3, x], {x, 2}] + 4*x*P[n - 2, x] - (2*x + 4*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + P[n - 1, x] + 2*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 + 4*x; v[4] = 1 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - 4*x*v[i - 3] + 4*x^2*deriv(v[i - 3], x) + 4*x^3*deriv(deriv(v[i - 3], x), x) + 4*x*v[i - 2] - (2*x + 4*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + v[i - 1] + 2*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)\u00b2"],"edges":["e-7389e9d417ab70c6"],"text":"A136011 \u2192 (\u03b8+1)\u00b2"},"headline":"A136011 \u2192 (\u03b8+1)\u00b2","chain":null}