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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":"71f45158bcfb55530417ee8ae433e008b08b707bb8bf4254088b0cbd66436f3a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eda3c556a6b4857f","role":"sequence","name":"Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","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: 0, 2;\n   3: 0, 2,    9;\n   4: 0, 2,   30,     60;\n   5: 0, 2,   75,    420,     525;\n   6: 0, 2,  168,   1960,    6300,     5670;\n   7: 0, 2,  357,   7672,   47250,   103950,     72765;\n   8: 0, 2,  738,  27300,  284900,  1143450,   1891890,   1081080;\n   9: 0, 2, 1503,  91740, 1514975,  9819810,  28693665,  37837800,  18243225;\n  10: 0, 2, 3036, 297264, 7436000, 72732660, 332552220, 756756000, 827026200, 344594250;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = ((2*n^2-8*n+8)*x-(n^3-7*n^2+16*n-12)*x^2)*P(n-2,x) + ((n-2)*x-(n^2-6*n+8)*x^2-(n^2-5*n+6)*x^3)*d/dx P(n-2,x) - (n-2-(2*n^2-8*n+6)*x)*P(n-1,x) + ((n-3)*x+(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2*x, P(3,x) = 2*x + 9*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[3, x_] := 2*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*P[n - 2, x] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] - (n - 2 - (2*n^2 - 8*n + 6)*x)*P[n - 1, x] + ((n - 3)*x + (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; v[3] = 2*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = (((2*n^2 - 8*n + 8)*x - (n^3 - 7*n^2 + 16*n - 12)*x^2)*v[i - 2] + ((n - 2)*x - (n^2 - 6*n + 8)*x^2 - (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) - (n - 2 - (2*n^2 - 8*n + 6)*x)*v[i - 1] + ((n - 3)*x + (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); 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-eda3c556a6b4857f","steps":["\u03b8+1"],"edges":["e-f8a67b70e7fd1d10"],"text":"A269939 \u2192 \u03b8+1"},"headline":"A269939 \u2192 \u03b8+1","chain":null}