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%I #13 Nov 03 2017 23:57:44
%S 1,1,4,34,480,9806,265810,9031672,369946976,17763808796,979237588724,
%T 60980447270606,4235264029208444,324666123698241278,
%U 27234289478093347800,2481803404797989770784,244176750407604670797512,25799371668963384500485336,2913817630601365750195653270,350339454794537924316740555298,44680403720910899340653296364594
%N a(n) = (-1)^n * A294359(n) / (n+1)^2.
%C It is conjectured that all terms are even after the initial '1,1'.
%C Sequence A294359(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = Sum_{n>=0} x^A003714(n) where A003714 is the Fibbinary numbers.
%C Fibbinary numbers are integers whose binary representation contains no consecutive ones (see A003714 for definition).
%H Paul D. Hanna, <a href="/A294475/b294475.txt">Table of n, a(n) for n = 0..600</a>
%F a(n) = (-1)^n * [x^n] F(x)^(-(n+1)^2) / (n+1)^2 such that F(x) = F(x^2) + x*F(x^4), where F(x) = Sum_{n>=0} x^A003714(n) and A003714 is the Fibbinary numbers.
%e Given the characteristic function of the Fibbinary numbers (A003714):
%e F(x) = 1 + x + x^2 + x^4 + x^5 + x^8 + x^9 + x^10 + x^16 + x^17 + x^18 + x^20 + x^21 + x^32 + x^33 + x^34 + x^36 + x^37 + x^40 + x^41 + x^42 + x^64 + x^65 + x^66 + x^68 + x^69 + x^72 + x^73 + x^74 + x^80 +...+ x^A003714(n) +...
%e such that F(x) = F(x^2) + x*F(x^4),
%e then this sequence equals the coefficients of x^n in F(x)^(-(n+1)^2).
%e ILLUSTRATION OF TERMS.
%e The table of coefficients of x^k in F(x)^(-n^2) begins:
%e n=1: [1, -1, 0, 1, -2, 1, 2, -4, 2, 3, -8, 7, 4, -16, 16, 2, -30, ...];
%e n=2: [1, -4, 6, 0, -19, 40, -26, -56, 166, -160, -110, 560, -705, ...];
%e n=3: [1, -9, 36, -75, 36, 279, -942, 1278, 531, -5956, 11700, ...];
%e n=4: [1, -16, 120, -544, 1548, -2192, -2720, 23936, -63426, 67984, ...];
%e n=5: [1, -25, 300, -2275, 12000, -45005, 112450, -116350, -441375, ...];
%e n=6: [1, -36, 630, -7104, 57573, -353016, 1668774, -5996664, ...];
%e n=7: [1, -49, 1176, -18375, 209426, -1846859, 13024690, -74680760, ...];
%e n=8: [1, -64, 2016, -41600, 631216, -7491392, 72180992, -578027008, ...]; ...
%e in which the main diagonal, divided by (-1)^n*(n+1)^2, forms this sequence:
%e A = [1, 4/2^2, 36/3^2, 544/4^2, 12000/5^2, 353016/6^2, 13024690/7^2, ...].
%e RELATED SERIES.
%e Define g.f. A(x) = Sum_{n>=0} a(n)*x^n, then
%e A(x) = 1 + x + 4*x^2 + 34*x^3 + 480*x^4 + 9806*x^5 + 265810*x^6 + 9031672*x^7 + 369946976*x^8 + 17763808796*x^9 + 979237588724*x^10 +...
%e and the logarithm of A(x) begins:
%e Log(A(x)) = x + 7*x^2/2 + 91*x^3/3 + 1767*x^4/4 + 46181*x^5/5 + 1525351*x^6/6 + 61073419*x^7/7 + 2878197983*x^8/8 + 156203074099*x^9/9 + 9599753499547*x^10/10 + 659252738855129*x^11/11 + 50047297413657447*x^12/12 +...
%e in which it appears that the coefficients consist entirely of odd integers.
%Y Cf. A294359, A003714.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 03 2017
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