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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":"d6ed1b5f84449bfcd30762536983f60c018e95b1405b595fe0d6bb9026635ca5","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-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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;\n  2:  5,   2;\n  3:  7,  10;\n  4:  9,  28,    3;\n  5: 11,  60,   21;\n  6: 13, 110,   81,   4;\n  7: 15, 182,  231,  36;\n  8: 17, 280,  546, 176,  5;\n  9: 19, 408, 1134, 624, 55;\n","recurrence_sentence":"The row polynomials satisfy (8*n-4-(4*n^3-2*n^2+6)*x+(n^3+2*n^2+n)*x^2)*P(n,x) = (8*n+4-(4*n^3+2*n^2-16*n+34)*x-(3*n^3-2*n^2-13*n)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) + (16*x-(8*n^2+16*n+20)*x^2+(10*n^2+22*n+8)*x^3-(2*n^2+6*n+4)*x^4)*d/dx 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[((8*n + 4 - (4*n^3 + 2*n^2 - 16*n + 34)*x - (3*n^3 - 2*n^2 - 13*n)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] + (16*x - (8*n^2 + 16*n + 20)*x^2 + (10*n^2 + 22*n + 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(8*n - 4 - (4*n^3 - 2*n^2 + 6)*x + (n^3 + 2*n^2 + n)*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] = ((8*n + 4 - (4*n^3 + 2*n^2 - 16*n + 34)*x - (3*n^3 - 2*n^2 - 13*n)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] + (16*x - (8*n^2 + 16*n + 20)*x^2 + (10*n^2 + 22*n + 8)*x^3 - (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(8*n - 4 - (4*n^3 - 2*n^2 + 6)*x + (n^3 + 2*n^2 + n)*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-d4c5acf24bc40277","steps":["\u03b8+1"],"edges":["e-25fc92be44d43b52"],"text":"A208509 \u2192 \u03b8+1"},"headline":"A208509 \u2192 \u03b8+1","chain":null}