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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":"6b5b2691c0a034d7ad0e31a7a94fcf80dd561d3e096d7417a87f9e14bd674ff5","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-c498b183c57d9f9c","role":"sequence","name":"veronese_section_2_0 of Triangular array binomial(2*n-k, k), k=0..n, n >= 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: 1;\n  2: 1,   2;\n  3: 1,  12;\n  4: 1,  30,    3;\n  5: 1,  56,   45;\n  6: 1,  90,  210,    4;\n  7: 1, 132,  630,  112;\n  8: 1, 182, 1485,  840,   5;\n  9: 1, 240, 3003, 3696, 225;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 12*x, P(4,x) = 1 + 30*x + 3*x^2, P(5,x) = 1 + 56*x + 45*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 1 + 12*x;\nP[4, x_] := 1 + 30*x + 3*x^2;\nP[5, x_] := 1 + 56*x + 45*x^2;\nP[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*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] = 1; v[3] = 1 + 2*x; v[4] = 1 + 12*x; v[5] = 1 + 30*x + 3*x^2; v[6] = 1 + 56*x + 45*x^2; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 15*x^2)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 3*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (27 + 10*x)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (11 - 3*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + 5*v[i - 1] - (4 + 4*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-6758fe7729845be8","steps":["Veronese 2","\u03b8+1"],"edges":["e-b36b946d9b6c256e","e-3185dff6e68ce282"],"text":"A054142 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A054142 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}