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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":"b77ca2dc3b8c5b73bd66add88ca68617ac00f3940a4e70dc4ffca60d89736698","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fdb186db0412847e","role":"sequence","name":"basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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:         3,         12;\n   3:        15,         96,         108;\n   4:       105,        852,        1620,         864;\n   5:       945,       8928,       22248,       20736,        6480;\n   6:     10395,     109548,      325080,      406080,      226800,       46656;\n   7:    135135,    1548864,     5211972,     7879680,     5994000,     2239488,      326592;\n   8:   2027025,   24854580,    92119356,   159890976,   149007600,    76749120,    20575296,    2239488;\n   9:  34459425,  446852160,  1789720272,  3455101440,  3732311520,  2377589760,   887023872,  179159040,   15116544;\n  10: 654729075, 8903704140, 38026354608, 79964689536, 96827179680, 72046848384, 33496581888, 9486471168, 1496537856, 100776960;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(24*n-36)*x+36*x^2)*P(n-2,x) + (4*n-4+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*P[n - 2, x] + (4*n - 4 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 12*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (24*n - 36)*x + 36*x^2)*v[i - 2] + (4*n - 4 + 12*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013610","\u03b8+1"],"edges":["e-1b98e7733a40a4e4","e-96ad5ca1ad701030"],"text":"A161198 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A161198 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}