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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":"365eb73284c752c2f56c9d7353af16d22582a0a11e4f9bdfced68df2f170f835","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-817434b322e3be5b","role":"sequence","name":"basis_map_A013609 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:      5,       4;\n  2:     22,      40,       12;\n  3:     95,     276,      180,       32;\n  4:    409,    1640,     1692,      640,       80;\n  5:   1760,    9008,    12840,     7616,     2000,      192;\n  6:   7573,   47160,    86148,    70400,    28800,     5760,     448;\n  7:  32585,  239068,   533700,   558880,   314000,    97344,   15680,   1024;\n  8: 140206, 1184200,  3127260,  4006400,  2880800,  1223040,  304192,  40960,   2304;\n  9: 603275, 5764620, 17587800, 26694720, 23478000, 12737088, 4327680, 897024, 103680, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(104*n+40*n*x-16*n^3*x^2)*P(n,x) = -(260*n+260+(52*n^2+308*n+256)*x-(40*n^3+40*n^2-80*n-80)*x^2-(16*n^3+32*n^2+16*n)*x^3)*P(n-1,x) - (169+390*x-(52*n^2+52*n-252)*x^2-(80*n^2+80*n-40)*x^3-(16*n^2+16*n)*x^4)*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[(-(260*n + 260 + (52*n^2 + 308*n + 256)*x - (40*n^3 + 40*n^2 - 80*n - 80)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*P[n - 1, x] - (169 + 390*x - (52*n^2 + 52*n - 252)*x^2 - (80*n^2 + 80*n - 40)*x^3 - (16*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(-104*n - 40*n*x + 16*n^3*x^2)]];\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] = (-(260*n + 260 + (52*n^2 + 308*n + 256)*x - (40*n^3 + 40*n^2 - 80*n - 80)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*v[i - 1] - (169 + 390*x - (52*n^2 + 52*n - 252)*x^2 - (80*n^2 + 80*n - 40)*x^3 - (16*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(-104*n - 40*n*x + 16*n^3*x^2)); 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-6466a8550c08a52f","steps":["basis A013609","\u03b8+1"],"edges":["e-0bad92788b2f6169","e-c6906127741b53c3"],"text":"A116414 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A116414 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}