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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":"be2d6d48a7d0452d25bc49d8c156d425e62360cc264a9752afcf011362b88fef","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-a679863398df5985","role":"sequence","name":"Triangle of coefficients in expansion of (14 + x)^n.","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:          14,           2;\n  2:         196,          56,           3;\n  3:        2744,        1176,         126,          4;\n  4:       38416,       21952,        3528,        224,         5;\n  5:      537824,      384160,       82320,       7840,       350,        6;\n  6:     7529536,     6453888,     1728720,     219520,     14700,      504,       7;\n  7:   105413504,   105413504,    33882912,    5378240,    480200,    24696,     686,     8;\n  8:  1475789056,  1686616064,   632481024,  120472576,  13445600,   921984,   38416,   896,    9;\n  9: 20661046784, 26564203008, 11384658432, 2529924096, 338829120, 29042496, 1613472, 56448, 1134, 10;\n","recurrence_sentence":"The row polynomials satisfy (14+n*x)*P(n,x) = (196+(14*n+28)*x+(n+1)*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[((196 + (14*n + 28)*x + (n + 1)*x^2)*P[n - 1, x])/(14 + 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] = ((196 + (14*n + 28)*x + (n + 1)*x^2)*v[i - 1])/(14 + 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-a679863398df5985","steps":["\u03b8+1"],"edges":["e-db3699374c20137e"],"text":"A147716 \u2192 \u03b8+1"},"headline":"A147716 \u2192 \u03b8+1","chain":null}