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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":"c4081555d2942dc20ed0fa74b9b934925ade8d3f83540dab97d593b6e505959a","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-b7cf68162d40a956","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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:        1;\n  1:        7,         22;\n  2:       49,        308,         363;\n  3:      343,       3234,        7623,        5324;\n  4:     2401,      30184,      106722,      149072,        73205;\n  5:    16807,     264110,     1245090,     2608760,      2562175,       966306;\n  6:   117649,    2218524,    13073445,    36522640,     53805675,     40584852,     12400927;\n  7:   823543,   18117946,   128119761,   447402340,    878826025,    994328874,    607645423,    155897368;\n  8:  5764801,  144943568,  1195784436,  5010906208,  12303564350,  18560805648,  17014071844,   8730252608,   1929229929;\n  9: 40353607, 1141430598, 10762059924, 52614515184, 155024910810, 292332688956, 357295508724, 275002957152, 121541485527, 23579476910;\n","recurrence_sentence":"The row polynomials satisfy (7+11*n*x)*P(n,x) = (49+(77*n+154)*x+(121*n+121)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((49 + (77*n + 154)*x + (121*n + 121)*x^2)*P[n - 1, x])/(7 + 11*n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((49 + (77*n + 154)*x + (121*n + 121)*x^2)*v[i - 1])/(7 + 11*n*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-b7cf68162d40a956","steps":["\u03b8+1"],"edges":["e-1841f6313ae4b6a3"],"text":"A038277 \u2192 \u03b8+1"},"headline":"A038277 \u2192 \u03b8+1","chain":null}