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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":"f4dde5d3ed466ca6255c10c36032e07ac72301ae49ffb90fbc49c4582c5cd815","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-170093e6e241015d","role":"sequence","name":"Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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:   2,       6;\n   3:   4,      42,        54;\n   4:   8,     234,       810,        648;\n   5:  16,    1218,      8586,      16848,        9720;\n   6:  32,    6186,     79650,     288360,      388800,       174960;\n   7:  64,   31122,    692874,    4127760,     9768600,      9972720,      3674160;\n   8: 128,  155994,   5823090,   53719848,   198676800,    345371040,    282910320,     88179840;\n   9: 256,  780738,  47988666,  660795408,  3587272920,   9447490080,  12910998240,   8817984000,   2380855680;\n  10: 512, 3905226, 390935970, 7844613480, 60133752000, 225178243920, 456617256480, 512677589760, 299987815680, 71425670400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+6*x)*P(n-1,x) + (3*x+3*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[(2 + 6*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], 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 + 6*x)*v[i - 1] + (3*x + 3*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-170093e6e241015d","steps":["\u03b8+1"],"edges":["e-cde218c4d4c37909"],"text":"A225472 \u2192 \u03b8+1"},"headline":"A225472 \u2192 \u03b8+1","chain":null}