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%I #13 Mar 06 2018 03:04:19
%S 1,1093,5461,205339,3897434,102033577,2464268044,63327787115,
%T 1627243839488,42592757748839,1123514934469218,29909548355299575,
%U 801531714260597080,21610508530971123358,585611144766061271010,15940294818101008311105,435592135387553867410170
%N a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^7, where Lucas(n) = A000204(n).
%H G. C. Greubel, <a href="/A203858/b203858.txt">Table of n, a(n) for n = 1..500</a>
%F G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^8 * x^n/n), which is the g.f. of A203808.
%F a(n) ~ phi^(7*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Sep 02 2017
%e G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^1093 * (1-4*x^3-x^6)^5461 * (1-7*x^4+x^8)^205339 * (1-11*x^5-x^10)^3897434 * (1-18*x^6+x^12)^102033577 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
%e where F(x) = exp( Sum_{n>=1} Lucas(n)^8 * x^n/n ) = g.f. of A203808:
%e F(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
%e where
%e log(F(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...
%t a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^7 &]/n; Array[a, 50] (* _G. C. Greubel_, Mar 05 2018 *)
%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^7)/n)}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=local(F=exp(sum(m=1, n, Lucas(m)^8*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}
%Y Cf. A203808, A203853, A203854, A203855, A203856, A203857, A203859, A203800.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 07 2012
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