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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":"ac3312b90fe503c1cbec351c5e722d15ab35c1a6f61380e9f210785d88c24ba4","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-587b2e7eeca5ddbb","role":"sequence","name":"basis_map_A013610 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,            48;\n  2:          361,          1824,           1728;\n  3:         6859,         51984,          98496,           55296;\n  4:       130321,       1316928,        3742848,         4202496,         1658880;\n  5:      2476099,      31277040,      118523520,       199618560,       157593600,        47775744;\n  6:     47045881,     713116512,     3377920320,      7585505280,      8982835200,      5446434816,      1337720832;\n  7:    893871739,   15807416016,    89852680512,    252218050560,    398239027200,    362187915264,    177916870656,     36691771392;\n  8:  16983563041,  343246747776,  2276267906304,   7667428737024,  15133083033600,  18350854373376,  13521682169856,   5577149251584,    990677827584;\n  9: 322687697779, 7336899233712, 55605973139712, 218521719005184, 517551439749120, 784499024461824, 770735883681792, 476846261010432, 169405908516864, 26418075402240;\n","recurrence_sentence":"The row polynomials satisfy (19+24*n*x)*P(n,x) = (361+(456*n+912)*x+(576*n+576)*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 + (456*n + 912)*x + (576*n + 576)*x^2)*P[n - 1, x])/(19 + 24*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 + (456*n + 912)*x + (576*n + 576)*x^2)*v[i - 1])/(19 + 24*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 A013610","\u03b8+1"],"edges":["e-f9ff39da4c6bf2a9","e-4979384718528f49"],"text":"A038322 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038322 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}