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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":"75c67c2f16b4876a7094fa6bb992e5b6897e71e85a93f77e0817931e2cc54c3c","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-5e3f8ad0b0a35d3e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","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,   10;\n  4: 1,   36,     15;\n  5: 1,  116,    183;\n  6: 1,  358,   1437,    244;\n  7: 1, 1086,   9333,   5540;\n  8: 1, 3272,  54810,  76112,   6925;\n  9: 1, 9832, 303498, 825104, 252605;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n+1+(2*n^2-2*n-4)*x-(n^3-n^2-2*n)*x^2)*P(n-2,x) + ((2*n+2)*x-(2*n^2+4*n+2)*x^2+(2*n^2+2*n)*x^3)*d/dx P(n-2,x) - (1-(2*n^2-2*n)*x)*P(n-1,x) + (2*n*x-2*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (2*n^2 - 2*n - 4)*x - (n^3 - n^2 - 2*n)*x^2)*P[n - 2, x] + ((2*n + 2)*x - (2*n^2 + 4*n + 2)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 2, x], x] - (1 - (2*n^2 - 2*n)*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + 1 + (2*n^2 - 2*n - 4)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 2] + ((2*n + 2)*x - (2*n^2 + 4*n + 2)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 2], x) - (1 - (2*n^2 - 2*n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n)); 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-d8312bb744732845","steps":["\u03b3 (scale 1/4)","\u03b8+1"],"edges":["e-4c46966ead0b055f","e-b88134e74c47445b"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 \u03b8+1"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 \u03b8+1","chain":null}