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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":"cd08cf707711940e6e0e22b551685e66f1f1119d3b9c199f13ce089d7343a99e","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-6bbd968fe5cbbe11","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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:     2,      2;\n  2:     5,     10,      3;\n  3:    14,     42,     27,      4;\n  4:    43,    172,    171,     56,      5;\n  5:   142,    710,    960,    500,    100,      6;\n  6:   499,   2994,   5130,   3700,   1200,    162,     7;\n  7:  1850,  12950,  26901,  24920,  11375,   2520,   245,    8;\n  8:  7193,  57544, 140574, 159488,  94675,  29904,  4802,  352,   9;\n  9: 29186, 262674, 738072, 992712, 728910, 302778, 69972, 8496, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*x+(6*n+12)*x^2+(n^2+3*n-2)*x^3-2*n*x^4)*P(n,x) = -(8*n-8+(6*n^2+12*n-42)*x+(n^3+8*n^2-17*n-52)*x^2+(n^3-2*n^2-25*n-14)*x^3-(3*n^2+n-4)*x^4+(2*n+2)*x^5)*P(n-1,x) + (8+(8*n+28)*x+(2*n^2+22*n+28)*x^2+(4*n^2+16*n)*x^3+(2*n^2-2*n-12)*x^4-(4*n+4)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 8 + (6*n^2 + 12*n - 42)*x + (n^3 + 8*n^2 - 17*n - 52)*x^2 + (n^3 - 2*n^2 - 25*n - 14)*x^3 - (3*n^2 + n - 4)*x^4 + (2*n + 2)*x^5)*P[n - 1, x] + (8 + (8*n + 28)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n^2 + 16*n)*x^3 + (2*n^2 - 2*n - 12)*x^4 - (4*n + 4)*x^5)*D[P[n - 1, x], x])/(12*x + (6*n + 12)*x^2 + (n^2 + 3*n - 2)*x^3 - 2*n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n - 8 + (6*n^2 + 12*n - 42)*x + (n^3 + 8*n^2 - 17*n - 52)*x^2 + (n^3 - 2*n^2 - 25*n - 14)*x^3 - (3*n^2 + n - 4)*x^4 + (2*n + 2)*x^5)*v[i - 1] + (8 + (8*n + 28)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n^2 + 16*n)*x^3 + (2*n^2 - 2*n - 12)*x^4 - (4*n + 4)*x^5)*deriv(v[i - 1], x))/(12*x + (6*n + 12)*x^2 + (n^2 + 3*n - 2)*x^3 - 2*n*x^4)); 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-07e5356ad37f61ea","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-2fc15f707287930e","e-662f5061022a36de"],"text":"A100862 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A100862 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}