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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":"8cd2c16004a15e955bafd26cdcb06940e3e07c9fca16c3ce736e4e33471561f2","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-649aa8398fb69113","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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:             361,             1064,               588;\n  1:            6859,            30324,             33516,             10976;\n  2:          130321,           768208,           1273608,            834176,            192080;\n  3:         2476099,         18244940,          40330920,          39623360,          18247600,           3226944;\n  4:        47045881,        415984632,        1149431220,        1505687680,        1040113200,         367871616,          52706752;\n  5:       893871739,       9220992676,       30574870452,       50064115360,       46111685200,       24463462464,        7009998016,         843308032;\n  6:     16983563041,     200227269536,      774563384784,     1521949106944,     1752244037600,     1239482098176,      532759849216,      128182820864,       13282101504;\n  7:    322687697779,    4279857886332,    18921476971152,    43375549547904,    59926746085920,    52987859697024,    30367311405312,    10959631183872,     2271239357184,     206610467840;\n  8:   6131066257801,   90352555378120,   449385078064860,  1177336344871680,  1897680292720800,  2013538668486912,  1442447291752320,   694109974978560,   215767738932480,   39255988889600,   3181801204736;\n  9: 116490258898219, 1888368407402708, 10435720146172860, 30757912009772640, 56659311596949600, 70138263618960768, 60294296795246976, 36267246192629760, 15031819145629440, 4102250838963200, 664996451789824, 48594782035968;\n","recurrence_sentence":"The row polynomials satisfy (19+(14*n+28)*x)*P(n,x) = (361+(266*n+1064)*x+(196*n+588)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1064*x + 588*x^2.","mathematica":"P[0, x_] := 361 + 1064*x + 588*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((361 + (266*n + 1064)*x + (196*n + 588)*x^2)*P[n - 1, x])/(19 + (14*n + 28)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1064*x + 588*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (266*n + 1064)*x + (196*n + 588)*x^2)*v[i - 1])/(19 + (14*n + 28)*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-5cf6acfdac607d6d","steps":["basis A013609","\u03b8+1"],"edges":["e-20f4338b8f3f0a8d","e-9e73d0f6469287bb"],"text":"A038333 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038333 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}