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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":"6dea1924d7fc99f90b468bb8064a84616f80b02b0e10b55366665f91bea5ecfa","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-a5324901f71bbafe","role":"sequence","name":"basis_map_A008277 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:          3,           2;\n  2:         18,          20,           3;\n  3:        162,         236,          63,          4;\n  4:       1944,        3392,        1236,        144,         5;\n  5:      29160,       58160,       26100,       4240,       275,        6;\n  6:     524880,     1163360,      609240,     123840,     11350,      468,       7;\n  7:   11022480,    26643680,    15757560,    3784480,    438550,    25788,     735,     8;\n  8:  264539520,   688136960,   449662080,  123191040,  17010000,  1274112,   52136,  1088,    9;\n  9: 7142567040, 19796913920, 14072405760, 4295280640, 681298800, 61183584, 3206952, 96576, 1539, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-21*n+12+(6*n-8)*x+x^2)*P(n-2,x) - ((3*n-4)*x+x^2)*d/dx P(n-2,x) + (6*n-4+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 21*n + 12 + (6*n - 8)*x + x^2)*P[n - 2, x] - ((3*n - 4)*x + x^2)*D[P[n - 2, x], x] + (6*n - 4 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 21*n + 12 + (6*n - 8)*x + x^2)*v[i - 2] - ((3*n - 4)*x + x^2)*deriv(v[i - 2], x) + (6*n - 4 + 2*x)*v[i - 1] + x*deriv(v[i - 1], 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-8229dbdd6a4b5cd9","steps":["basis A008277","\u03b8+1"],"edges":["e-30984bb1804f505f","e-48680ae9f15c8512"],"text":"A225470 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A225470 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}