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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":"4691c132986b5e43df6bb9f83a4e70310e1892c49c36ad6df5d6cbfdffeb8c43","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-4b2df01c2184142e","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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:            324,             2016,              2352;\n  1:           5832,            54432,            127008,              87808;\n  2:         104976,          1306368,           4572288,            6322176,            3073280;\n  3:        1889568,         29393280,         137168640,          284497920,          276595200,           103262208;\n  4:       34012224,        634894848,        3703553280,        10241925120,        14936140800,         11152318464,          3373232128;\n  5:      612220032,      13332791808,       93329542656,       322620641280,       627317913600,        702596063232,        425027248128,        107943428096;\n  6:    11019960576,     274274574336,     2239909023744,      9291474468864,     22583444889600,      33724611035136,      30601961865216,      15543853645824,       3400217985024;\n  7:   198359290368,    5554060130304,    51837894549504,    250869810659328,    731703614423040,    1365846746923008,    1652505940721664,    1259052145311744,     550835313573888,     105784559534080;\n  8:  3570467226624,  111081202606080,  1166352627363840,   6450937988382720,  21951108432691200,   49170482889228288,   74362767332474880,   75543128718704640,   49575178221649920,   19041220716134400,   3258164433649664;\n  9: 64268410079232, 2199407811600384, 25659757802004480, 159660715212472320, 620902781381836800, 1622625935344533504, 2944765586366005248, 3739384871575879680, 3271961762628894720, 1885080850897305600, 645116557862633472, 99522113609662464;\n","recurrence_sentence":"The row polynomials satisfy (9+(14*n+28)*x)*P(n,x) = (162+(252*n+1008)*x+(392*n+1176)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 2016*x + 2352*x^2.","mathematica":"P[0, x_] := 324 + 2016*x + 2352*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*P[n - 1, x])/(9 + (14*n + 28)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 2016*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*v[i - 1])/(9 + (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-7dcf11eddf2b9a8a","steps":["basis A013611","\u03b8+1"],"edges":["e-e460f89b751823ed","e-7dbf73acc9538375"],"text":"A038321 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038321 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}