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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":"6e77d53f358e8dca595ef415fd95b876de21fcda5e219d419c409b67d70956c6","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, 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: 0,          2;\n  2: 0,          2,          3;\n  3: 0,          8,         15,          4;\n  4: 0,         72,        147,         56,         5;\n  5: 0,       1152,       2460,       1092,       150,        6;\n  6: 0,      28800,      63228,      30580,      5115,      330,       7;\n  7: 0,    1036800,    2319408,    1185184,    222365,    18018,     637,      8;\n  8: 0,   50803200,  115206192,   61166560,  12377365,  1149720,   52234,   1120,    9;\n  9: 0, 3251404800, 7449401088, 4068268096, 868609560, 88434918, 4684316, 131376, 1836, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(2*n^2-8*n+8)*x+x^2)*P(n-2,x) + (2*n^2-6*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*P[n - 2, x] + (2*n^2 - 6*n + 5 + 2*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*v[i - 2] + (2*n^2 - 6*n + 5 + 2*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-54284d24a094e2d3","steps":["\u03b8+1"],"edges":["e-b6a50d5d47e168f5"],"text":"A269944 \u2192 \u03b8+1"},"headline":"A269944 \u2192 \u03b8+1","chain":null}