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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":"47cab7f10a52c976cfcb968f8d7e6ce640c7f79b6a974ffd881ae08005e3bf96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9e4edf560ed848c1","role":"sequence","name":"Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0,      2;\n   3: 0,      4,       3;\n   4: 0,     12,      18,       4;\n   5: 0,     48,     108,      48,       5;\n   6: 0,    240,     720,     480,     100,       6;\n   7: 0,   1440,    5400,    4800,    1500,     180,      7;\n   8: 0,  10080,   45360,   50400,   21000,    3780,    294,     8;\n   9: 0,  80640,  423360,  564480,  294000,   70560,   8232,   448,   9;\n  10: 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10;\n","recurrence_sentence":"The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(2*n-4-(n^2-3*n-2)*x-n*x^2)*P(n-1,x) - (2*x-n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 - (n^2 - 3*n - 2)*x - n*x^2)*P[n - 1, x] - (2*x - n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n - 4 - (n^2 - 3*n - 2)*x - n*x^2)*v[i - 1] - (2*x - n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9e4edf560ed848c1","steps":["\u03b8+1"],"edges":["e-8a6ae03a068873bd"],"text":"A271703 \u2192 \u03b8+1"},"headline":"A271703 \u2192 \u03b8+1","chain":null}