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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":"e8e42773724c9abb83c6f4f92d923a53114d77e596a990d5d05c393d9c2765e7","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-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:           19,            16;\n  2:          361,           608,           192;\n  3:         6859,         17328,         10944,          2048;\n  4:       130321,        438976,        415872,        155648,         20480;\n  5:      2476099,      10425680,      13169280,       7393280,       1945600,        196608;\n  6:     47045881,     237705504,     375324480,     280944640,     110899200,      22413312,       1835008;\n  7:    893871739,    5269138672,    9983631168,    9341409280,    4916531200,    1490485248,     244056064,     16777216;\n  8:  16983563041,  114415582592,  252918656256,  283978842112,  186828185600,   75517919232,   18548260864,   2550136832,   150994944;\n  9: 322687697779, 2445633077904, 6178441459968, 8093397000192, 6389523947520, 3228391047168, 1057250869248, 218036699136, 25820135424, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy (19+8*n*x)*P(n,x) = (361+(152*n+304)*x+(64*n+64)*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[((361 + (152*n + 304)*x + (64*n + 64)*x^2)*P[n - 1, x])/(19 + 8*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] = ((361 + (152*n + 304)*x + (64*n + 64)*x^2)*v[i - 1])/(19 + 8*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-2f02b2035f8f4541","steps":["basis A007318","\u03b8+1"],"edges":["e-c22af348c5e552a2","e-5c4a6723917ce0da"],"text":"A038322 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038322 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}