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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":"d4503f45b5e0733e35047ac8e496b687c7595d7c60bcb7b6fa199975167e850f","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-3afe7d5f29fd0242","role":"sequence","name":"basis_map_A013611 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,             2128,              2352;\n  1:            6859,            60648,            134064,              87808;\n  2:          130321,          1536416,           5094432,            6673408,            3073280;\n  3:         2476099,         36489880,         161323680,          316986880,          291961600,           103262208;\n  4:        47045881,        831969264,        4597724880,        12045501440,        16641811200,         11771891712,          3373232128;\n  5:       893871739,      18441985352,      122299481808,       400512922880,       737786963200,        782830798848,        448639873024,        107943428096;\n  6:     16983563041,     400454539072,     3098253539136,     12175592855552,     28035904601600,      39663427141632,      34096630349824,      16407401070592,       3400217985024;\n  7:    322687697779,    8559715772664,    75685907884608,    347004396383232,    958827937374720,    1695611510304768,    1943507929939968,    1402832791535616,     581437275439104,     105784559534080;\n  8:   6131066257801,  180705110756240,  1797540312259440,   9418690758973440,  30362884683532800,   64433237391581184,   92316626672148480,   88846076797255680,   55236541166714880,   20099066311475200,   3258164433649664;\n  9: 116490258898219, 3776736814805416, 41742880584691440, 246063296078181120, 906548985551193600, 2244424435806744576, 3858834994895806464, 4642207512656609280, 3848145701281136640, 2100352429549158400, 680956366632779776, 99522113609662464;\n","recurrence_sentence":"The row polynomials satisfy (19+(28*n+56)*x)*P(n,x) = (361+(532*n+2128)*x+(784*n+2352)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 2128*x + 2352*x^2.","mathematica":"P[0, x_] := 361 + 2128*x + 2352*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*P[n - 1, x])/(19 + (28*n + 56)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 2128*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (532*n + 2128)*x + (784*n + 2352)*x^2)*v[i - 1])/(19 + (28*n + 56)*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 A013611","\u03b8+1"],"edges":["e-0d83dd77e9f6bf78","e-fd35dbb0c2fd1687"],"text":"A038333 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038333 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}