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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":"e2ab24766c1398507e778e0fcf29491efeaae2016a4b9a8b5b477bb1f5863819","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-89f02ea58ff2bc55","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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:            23,             24;\n  2:           529,           1104,            432;\n  3:         12167,          38088,          29808,           6912;\n  4:        279841,        1168032,        1371168,         635904,         103680;\n  5:       6436343,       33580920,       52561440,       36564480,       11923200,        1492992;\n  6:     148035889,      926833392,     1813369680,     1681966080,      822700800,      206032896,       20901888;\n  7:    3404825447,    24870029352,    58390503696,    67699134720,    44151609600,    16585648128,     3365203968,     286654464;\n  8:   78310985281,   653726485824,  1790642113344,  2491328157696,  2030974041600,  1017253085184,   309598765056,   52744421376,   3869835264;\n  9: 1801152661463, 16915172820696, 52951845351744, 85950821440512, 84082325322240, 52642847158272, 21362314788864, 5459047612416, 801055899648, 51597803520;\n","recurrence_sentence":"The row polynomials satisfy (23+12*n*x)*P(n,x) = (529+(276*n+552)*x+(144*n+144)*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[((529 + (276*n + 552)*x + (144*n + 144)*x^2)*P[n - 1, x])/(23 + 12*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] = ((529 + (276*n + 552)*x + (144*n + 144)*x^2)*v[i - 1])/(23 + 12*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-9e472291b944d56e","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-37d2e50d1bfe4331","e-ba8ec891d4c028d8"],"text":"A038326 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038326 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}