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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":"a9893ec3440d9a5ee0bddd0722051088fdeffba68e965a785d36b83544d0b188","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-e7e87c6a9f61cef2","role":"sequence","name":"veronese_section_2_0 of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","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:  2;\n  2:  3,   2;\n  3:  4,  12;\n  4:  5,  36,    3;\n  5:  6,  80,   30;\n  6:  7, 150,  135,    4;\n  7:  8, 252,  420,   56;\n  8:  9, 392, 1050,  336,  5;\n  9: 10, 576, 2268, 1344, 90;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-3*x+3*x^2-x^3)*P(n-6,x) + (6-12*x+6*x^2)*P(n-5,x) - (15-18*x+3*x^2)*P(n-4,x) + (20-12*x)*P(n-3,x) - (15-3*x)*P(n-2,x) + 6*P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 3 + 2*x, P(3,x) = 4 + 12*x, P(4,x) = 5 + 36*x + 3*x^2, P(5,x) = 6 + 80*x + 30*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 3 + 2*x;\nP[3, x_] := 4 + 12*x;\nP[4, x_] := 5 + 36*x + 3*x^2;\nP[5, x_] := 6 + 80*x + 30*x^2;\nP[n_, x_] := P[n, x] = Expand[-(1 - 3*x + 3*x^2 - x^3)*P[n - 6, x] + (6 - 12*x + 6*x^2)*P[n - 5, x] - (15 - 18*x + 3*x^2)*P[n - 4, x] + (20 - 12*x)*P[n - 3, x] - (15 - 3*x)*P[n - 2, x] + 6*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] = 2; v[3] = 3 + 2*x; v[4] = 4 + 12*x; v[5] = 5 + 36*x + 3*x^2; v[6] = 6 + 80*x + 30*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(1 - 3*x + 3*x^2 - x^3)*v[i - 6] + (6 - 12*x + 6*x^2)*v[i - 5] - (15 - 18*x + 3*x^2)*v[i - 4] + (20 - 12*x)*v[i - 3] - (15 - 3*x)*v[i - 2] + 6*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-f24dc5f77f658331","steps":["Veronese 2","\u03b8+1"],"edges":["e-582e2f23d54a43e5","e-f2bc5e4256ec08f7"],"text":"A093375 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A093375 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}