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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":"67e800dc43442b0a369bb43b9a6634a939538ad662c8bd5a062bcc1d8c41c6e2","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-9c5c208df8605708","role":"sequence","name":"Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:           3,           2;\n  2:          21,          20,           3;\n  3:         231,         262,          63,          4;\n  4:        3465,        4392,        1338,        144,          5;\n  5:       65835,       90378,       32010,       4520,        275,        6;\n  6:     1514205,     2210364,      871797,     146640,      11975,      468,       7;\n  7:    40883535,    62708238,    26854065,    5121676,     506625,    27006,     735,      8;\n  8:  1267389585,  2025722448,   926538372,  194577376,   22107470,  1445136,   54292,   1088,    9;\n  9: 44358635475, 73435064850, 35467426692, 8045592656, 1016983170, 77108724, 3586212, 100128, 1539, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-40*n+25+(8*n-10)*x+x^2)*P(n-2,x) + (8*n-6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 40*n + 25 + (8*n - 10)*x + x^2)*P[n - 2, x] + (8*n - 6 + 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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 40*n + 25 + (8*n - 10)*x + x^2)*v[i - 2] + (8*n - 6 + 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-9c5c208df8605708","steps":["\u03b8+1"],"edges":["e-4baba0677ba64ec6"],"text":"A225471 \u2192 \u03b8+1"},"headline":"A225471 \u2192 \u03b8+1","chain":null}