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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":"e994b157f5557342a3f3e48afe6e8eca1d8278f302e8bdc797b856144a42d3de","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-e224ad3e3147ef52","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.","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:      4,         8;\n  2:     16,       112,          48;\n  3:     64,      1104,        1656,          256;\n  4:    256,      9392,       33120,        18784,         1280;\n  5:   1024,     73936,      514344,       685792,       184840,         6144;\n  6:   4096,    556464,     6917904,     17830592,     11529840,      1669392,        28672;\n  7:  16384,   4075472,    85038264,    379379648,    474224560,    170076528,     14264152,      131072;\n  8:  65536,  29347504,   984873216,   7086238112,  15175185920,  10629357168,   2298037504,   117390016,    589824;\n  9: 262144, 209102544, 10949263560, 121010709792, 411101129880, 493321355856, 211768742136, 29198036160, 940961448, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (3*n+1)*P(n,x) = -(12*n+16+(18*n^2+12*n-16)*x+(27*n^3+27*n^2-18*n-8)*x^2)*P(n-2,x) - ((9*n+12)*x+(27*n^2+36*n)*x^2-(27*n^2+45*n+12)*x^3)*d/dx P(n-2,x) + (15*n+8+(18*n^2+12*n+2)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*P[n - 2, x] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*D[P[n - 2, x], x] + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x])/(3*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*v[i - 2] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*deriv(v[i - 2], x) + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x))/(3*n + 1)); 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-e224ad3e3147ef52","steps":["\u03b8+1"],"edges":["e-a94914357edceb4c"],"text":"A257622 \u2192 \u03b8+1"},"headline":"A257622 \u2192 \u03b8+1","chain":null}