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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":"59d1da62e8199bd9049cc96f71b96356fb5be60606cf913150c9b38987f26b44","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-4479b0484a11b442","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","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:     2,      2;\n   3:     5,     12,       3;\n   4:    14,     56,      36,       4;\n   5:    42,    240,     270,      80,       5;\n   6:   132,    990,    1650,     880,     150,      6;\n   7:   429,   4004,    9009,    7280,    2275,    252,      7;\n   8:  1430,  16016,   45864,   50960,   24500,   5040,    392,     8;\n   9:  4862,  63648,  222768,  319872,  214200,  68544,   9996,   576,   9;\n  10: 16796, 251940, 1046520, 1860480, 1627920, 732564, 167580, 18240, 810, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-2*n^2-2*n+(4*n^4-2*n^3-2*n^2)*x)*P(n,x) = (14*n^3-22*n^2+8*n+(15*n^4-2*n^3-43*n^2+38*n-8)*x+(12*n^4-12*n^3-10*n^2+14*n-4)*x^2)*P(n-1,x) + (n^2-n+(n^3+5*n^2-8*n+2)*x+(7*n^3-4*n^2+3*n)*x^2-(8*n^3+2*n^2-6*n+2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n^3 - 22*n^2 + 8*n + (15*n^4 - 2*n^3 - 43*n^2 + 38*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*P[n - 1, x] + (n^2 - n + (n^3 + 5*n^2 - 8*n + 2)*x + (7*n^3 - 4*n^2 + 3*n)*x^2 - (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((14*n^3 - 22*n^2 + 8*n + (15*n^4 - 2*n^3 - 43*n^2 + 38*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*v[i - 1] + (n^2 - n + (n^3 + 5*n^2 - 8*n + 2)*x + (7*n^3 - 4*n^2 + 3*n)*x^2 - (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)); 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-4479b0484a11b442","steps":["\u03b8+1"],"edges":["e-ddf41d3a97c5fd4f"],"text":"A120986 \u2192 \u03b8+1"},"headline":"A120986 \u2192 \u03b8+1","chain":null}