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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"bbf08cccda4ff62333b9c66eff2cd3301a768b98913f4581e2d84be5335fe83c","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","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:            289,             408,             108;\n  1:           4913,           10404,            5508,             864;\n  2:          83521,          235824,          187272,           58752,            6480;\n  3:        1419857,         5011260,         5306040,         2496960,          550800,           46656;\n  4:       24137569,       102229704,       135304020,        84896640,        28090800,         4758912,          326592;\n  5:      410338673,      2027555796,      3220235676,      2525675040,      1114268400,       283155264,        38864448,        2239488;\n  6:     6975757441,     39392512608,     72992008656,     68698361088,     37885125600,     12836371968,      2642782464,      304570368,       15116544;\n  7:   118587876497,    753381803628,   1595396760624,   1751808207744,   1159284843360,    490991227776,    134781905664,    23299633152,     2312831232,     100776960;\n  8:  2015993900449,  14230545179640,  33902181163260,  42543913616640,  32846403895200,  16693701744384,   5728230990720,  1320312545280,   196590654720,   17132083200,    665127936;\n  9: 34271896307633, 266111194859268, 704411986392180, 994463980788960, 877468218343200, 520287037699968, 214235839052928, 61724611491840, 12254150810880, 1601849779200, 124378924032, 4353564672;\n","recurrence_sentence":"The row polynomials satisfy (17+(6*n+12)*x)*P(n,x) = (289+(102*n+408)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 408*x + 108*x^2.","mathematica":"P[0, x_] := 289 + 408*x + 108*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (102*n + 408)*x + (36*n + 108)*x^2)*P[n - 1, x])/(17 + (6*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 408*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (102*n + 408)*x + (36*n + 108)*x^2)*v[i - 1])/(17 + (6*n + 12)*x)); 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-1f2dec04776afb2b","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-653ba39db5056185","e-1c5ba3af81158131"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}