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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":"457acdb47e332a8cb3e42292baed50f718efeea30fb3b79aff309e97852509bc","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,           4;\n  2:         196,         112,           9;\n  3:        2744,        2352,         378,          16;\n  4:       38416,       43904,       10584,         896,         25;\n  5:      537824,      768320,      246960,       31360,       1750,        36;\n  6:     7529536,    12907776,     5186160,      878080,      73500,      3024,       49;\n  7:   105413504,   210827008,   101648736,    21512960,    2401000,    148176,     4802,     64;\n  8:  1475789056,  3373232128,  1897443072,   481890304,   67228000,   5531904,   268912,   7168,    81;\n  9: 20661046784, 53128406016, 34153975296, 10119696384, 1694145600, 174254976, 11294304, 451584, 10206, 100;\n","recurrence_sentence":"The row polynomials satisfy (196+(42*n-14)*x+n^2*x^2)*P(n,x) = (2744+(588*n+588)*x+(14*n^2+70*n+42)*x^2+(n^2+2*n+1)*x^3)*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[((2744 + (588*n + 588)*x + (14*n^2 + 70*n + 42)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(196 + (42*n - 14)*x + n^2*x^2)]];\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] = ((2744 + (588*n + 588)*x + (14*n^2 + 70*n + 42)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(196 + (42*n - 14)*x + n^2*x^2)); 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)\u00b2"],"edges":["e-8d6c7cc2c58bb00b"],"text":"A147716 \u2192 (\u03b8+1)\u00b2"},"headline":"A147716 \u2192 (\u03b8+1)\u00b2","chain":null}