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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":"cc7201eeeb737459891a721dc10c5fb2b88297d220e412cac2d75a9411c4e22a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   2,       2;\n   3:   4,      26,         3;\n   4:   8,     186,       180,          4;\n   5:  16,    1090,      3393,       1004,           5;\n   6:  32,    5866,     43494,      42712,        5090,           6;\n   7:  64,   30354,    462339,    1049752,      444990,       24534,          7;\n   8: 128,  154202,   4427208,   19561148,    19351760,     4154994,     114632,         8;\n   9: 256,  776642,  39812421,  309619732,   588793295,   302856018,   36225091,    524216,       9;\n  10: 512, 3896010, 344128554, 4414398384, 14435500770, 14333727708, 4257822030, 301857696, 2359206, 10;\n","recurrence_sentence":"The row polynomials satisfy (3*n-5)*P(n,x) = (6*n-4+(18*n^2-60*n+32)*x-(27*n^3-135*n^2+198*n-80)*x^2)*P(n-2,x) + ((9*n-6)*x-(27*n^2-54*n+24)*x^2+(27*n^2-63*n+30)*x^3)*d/dx P(n-2,x) + (3*n-8+(18*n^2-60*n+50)*x)*P(n-1,x) + ((9*n-15)*x-(9*n-15)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n - 4 + (18*n^2 - 60*n + 32)*x - (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*P[n - 2, x] + ((9*n - 6)*x - (27*n^2 - 54*n + 24)*x^2 + (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], x] + (3*n - 8 + (18*n^2 - 60*n + 50)*x)*P[n - 1, x] + ((9*n - 15)*x - (9*n - 15)*x^2)*D[P[n - 1, x], x])/(3*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((6*n - 4 + (18*n^2 - 60*n + 32)*x - (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*v[i - 2] + ((9*n - 6)*x - (27*n^2 - 54*n + 24)*x^2 + (27*n^2 - 63*n + 30)*x^3)*deriv(v[i - 2], x) + (3*n - 8 + (18*n^2 - 60*n + 50)*x)*v[i - 1] + ((9*n - 15)*x - (9*n - 15)*x^2)*deriv(v[i - 1], x))/(3*n - 5)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-daa3b3e1ead00edc","steps":["\u03b8+1"],"edges":["e-d1288c2765fb6613"],"text":"A225117 \u2192 \u03b8+1"},"headline":"A225117 \u2192 \u03b8+1","chain":null}