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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":"5c70d829cff4857f1bd902cc24411876d143cd7f0f0670bcb8677b38fdf7d883","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-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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:  1,   2;\n   3:  2,   6,    3;\n   4:  3,  14,   15,    4;\n   5:  5,  30,   48,   28,    5;\n   6:  8,  60,  129,  116,   45,    6;\n   7: 13, 116,  312,  380,  230,   66,    7;\n   8: 21, 218,  705, 1084,  895,  402,   91,   8;\n   9: 34, 402, 1518, 2820, 2955, 1818,  644, 120,   9;\n  10: 55, 730, 3153, 6868, 8730, 6840, 3325, 968, 153, 10;\n","recurrence_sentence":"The row polynomials satisfy (10*n-10+(5*n^2+3*n-8)*x+(n^3+n^2-2*n)*x^2)*P(n,x) = -(20*n-50+(15*n^2-29*n-40)*x+(2*n^3-n^2-25*n-6)*x^2-(2*n^2+4*n)*x^3)*P(n-1,x) + (25+(25*n+45)*x+(5*n^2+40*n+23)*x^2+(6*n^2+17*n+3)*x^3+(n^2+2*n)*x^4)*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[(-(20*n - 50 + (15*n^2 - 29*n - 40)*x + (2*n^3 - n^2 - 25*n - 6)*x^2 - (2*n^2 + 4*n)*x^3)*P[n - 1, x] + (25 + (25*n + 45)*x + (5*n^2 + 40*n + 23)*x^2 + (6*n^2 + 17*n + 3)*x^3 + (n^2 + 2*n)*x^4)*D[P[n - 1, x], x])/(10*n - 10 + (5*n^2 + 3*n - 8)*x + (n^3 + n^2 - 2*n)*x^2)]];\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] = (-(20*n - 50 + (15*n^2 - 29*n - 40)*x + (2*n^3 - n^2 - 25*n - 6)*x^2 - (2*n^2 + 4*n)*x^3)*v[i - 1] + (25 + (25*n + 45)*x + (5*n^2 + 40*n + 23)*x^2 + (6*n^2 + 17*n + 3)*x^3 + (n^2 + 2*n)*x^4)*deriv(v[i - 1], x))/(10*n - 10 + (5*n^2 + 3*n - 8)*x + (n^3 + n^2 - 2*n)*x^2)); 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-c72653d9ab0ee8be","steps":["\u03b8+1"],"edges":["e-bf6c3a69738a7962"],"text":"A063967 \u2192 \u03b8+1"},"headline":"A063967 \u2192 \u03b8+1","chain":null}