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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":"88b9e93e6e7301ab74f06219ec7c64251d7ee87ef9131c1f4597ec3a9f58383d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-bf8bba6616804a77","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:      1,       2;\n  2:      2,       6;\n  3:      6,      20,      3;\n  4:     24,      84,     21;\n  5:    120,     432,    135,     4;\n  6:    720,    2640,    936,    52;\n  7:   5040,   18720,   7200,   544,    5;\n  8:  40320,  151200,  61560,  5600,  105;\n  9: 362880, 1370880, 582120, 60000, 1625, 6;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-4)*x*P(n-3,x) - (n^2-3*n+2-2*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 2 + 6*x, P(3,x) = 6 + 20*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 2 + 6*x;\nP[3, x_] := 6 + 20*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 4)*x*P[n - 3, x] - (n^2 - 3*n + 2 - 2*x)*P[n - 2, x] + (2*n - 2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 2 + 6*x; v[4] = 6 + 20*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 4)*x*v[i - 3] - (n^2 - 3*n + 2 - 2*x)*v[i - 2] + (2*n - 2)*v[i - 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-bf8bba6616804a77","steps":["\u03b8+1"],"edges":["e-ba6e3c63ca74b92f"],"text":"A249247 \u2192 \u03b8+1"},"headline":"A249247 \u2192 \u03b8+1","chain":null}