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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":"cc044156f9a1be9c368662136c5fe0f8ca0320ff4353c04bcc794216fd90de25","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-688fdf685007c036","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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:            289,             680,              300;\n  1:           4913,           17340,            15300,             4000;\n  2:          83521,          393040,           520200,           272000,            50000;\n  3:        1419857,         8352100,         14739000,         11560000,          4250000,           600000;\n  4:       24137569,       170382840,        375844500,        393040000,        216750000,         61200000,          7000000;\n  5:      410338673,      3379259660,       8945099100,      11692940000,       8597750000,       3641400000,        833000000,         80000000;\n  6:     6975757441,     65654187680,     202755579600,     318047968000,     292323500000,     165076800000,      56644000000,      10880000000,       900000000;\n  7:   118587876497,   1255636339380,    4431657668400,    8110223184000,    8945099100000,    6314187600000,    2888844000000,     832320000000,    137700000000,     10000000000;\n  8:  2015993900449,  23717575299400,   94172725453500,  196962563040000,  253444474500000,  214682378400000,  122775870000000,   47164800000000,  11704500000000,   1700000000000,   110000000000;\n  9: 34271896307633, 443518658098780, 1956699962200500, 4603999911060000, 6770588104500000, 6690934126800000, 4591817538000000, 2204954400000000, 729580500000000, 158950000000000, 20570000000000, 1200000000000;\n","recurrence_sentence":"The row polynomials satisfy (17+(10*n+20)*x)*P(n,x) = (289+(170*n+680)*x+(100*n+300)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 300*x^2.","mathematica":"P[0, x_] := 289 + 680*x + 300*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (170*n + 680)*x + (100*n + 300)*x^2)*P[n - 1, x])/(17 + (10*n + 20)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 680*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (170*n + 680)*x + (100*n + 300)*x^2)*v[i - 1])/(17 + (10*n + 20)*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-a5302200d285579d","steps":["basis A013609","\u03b8+1"],"edges":["e-2f341f062a4ee4be","e-7a182279327f6daa"],"text":"A038331 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038331 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}