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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":"c611611da5eb31e5db13f1cf9d4e5f6d02e9674c244bc8dcd65f951299d5949f","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-5d7c3f4772a9d8bb","role":"sequence","name":"shift_x_plus_one of 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:      2,       2;\n  3:      6,      10;\n  4:     24,      56,      15;\n  5:    120,     360,     183;\n  6:    720,    2640,    1986,     244;\n  7:   5040,   21840,   21798,    5540;\n  8:  40320,  201600,  250992,   98272,   6925;\n  9: 362880, 2056320, 3069360, 1633440, 252605;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+n^2-n-1+(2*n^3-6*n-4)*x+(n^3-n^2-2*n)*x^2)*P(n-2,x) + ((2*n^2+4*n+2)*x+(4*n^2+6*n+2)*x^2+(2*n^2+2*n)*x^3)*d/dx P(n-2,x) + (2*n^2+2*n+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^3 + n^2 - n - 1 + (2*n^3 - 6*n - 4)*x + (n^3 - n^2 - 2*n)*x^2)*P[n - 2, x] + ((2*n^2 + 4*n + 2)*x + (4*n^2 + 6*n + 2)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 2*n + 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^3 + n^2 - n - 1 + (2*n^3 - 6*n - 4)*x + (n^3 - n^2 - 2*n)*x^2)*v[i - 2] + ((2*n^2 + 4*n + 2)*x + (4*n^2 + 6*n + 2)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 2*n + 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)","x \u21a6 x+1","\u03b8+1"],"edges":["e-16125ab4d60fcfd7","e-42ac1b21fb8d0456","e-b821d3224ccf1e49"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}