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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":"7c543e9f0e0a4cacbc6f7f14ce303fb6cf7ba1cc4cf9416497ec1085be41608e","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-53dc6d1eeb1907f4","role":"sequence","name":"basis_map_A013609 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,             1008,              588;\n  1:           5832,            27216,            31752,             10976;\n  2:         104976,           653184,          1143072,            790272,            192080;\n  3:        1889568,         14696640,         34292160,          35562240,          17287200,           3226944;\n  4:       34012224,        317447424,        925888320,        1280240640,         933508800,         348509952,          52706752;\n  5:      612220032,       6666395904,      23332385664,       40327580160,       39207369600,       21956126976,        6641050752,         843308032;\n  6:    11019960576,     137137287168,     559977255936,     1161434308608,     1411465305600,     1053894094848,      478155654144,      121436356608,       13282101504;\n  7:   198359290368,    2777030065152,   12959473637376,    31358726332416,    45731475901440,    42682710841344,    25820405323776,     9836344885248,     2151700443648,     206610467840;\n  8:  3570467226624,   55540601303040,  291588156840960,   806367248547840,  1371944277043200,  1536577590288384,  1161918239569920,   590180693114880,   193653039928320,   37189884211200,   3181801204736;\n  9: 64268410079232, 1099703905800192, 6414939450501120, 19957589401559040, 38806423836364800, 50707060479516672, 46011962286968832, 29213944309186560, 12781100635269120, 3681798536908800, 629996638537728, 48594782035968;\n","recurrence_sentence":"The row polynomials satisfy (9+(7*n+14)*x)*P(n,x) = (162+(126*n+504)*x+(98*n+294)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 588*x^2.","mathematica":"P[0, x_] := 324 + 1008*x + 588*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((162 + (126*n + 504)*x + (98*n + 294)*x^2)*P[n - 1, x])/(9 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1008*x + 588*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((162 + (126*n + 504)*x + (98*n + 294)*x^2)*v[i - 1])/(9 + (7*n + 14)*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 A013609","\u03b8+1"],"edges":["e-322a2e4c0bbd3231","e-aa24980847f0bb3e"],"text":"A038321 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038321 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}