|
%I #20 Dec 11 2017 05:06:31
%S 1,364,1365,29230,354312,5667900,84974760,1347387210,21411102720,
%T 346282421940,5645803690800,92886793449030,1538448587832240,
%U 25635241395476100,429333683845968552,7222607529064709670,121980435560782376760,2067248664062116147200
%N a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).
%H G. C. Greubel, <a href="/A203857/b203857.txt">Table of n, a(n) for n = 1..795</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)^7 * x^n/n), which is the g.f. of A203807.
%F a(n) ~ phi^(6*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)^364 * (1-4*x^3-x^6)^1365 * (1-7*x^4+x^8)^29230 * (1-11*x^5-x^10)^354312 * (1-18*x^6+x^12)^5667900 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
%e where F(x) = exp( Sum_{n>=1} Lucas(n)^7 * x^n/n ) = g.f. of A203807:
%e F(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
%e where log(F(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...
%t a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^6&]/n; Array[a, 18] (* _Jean-François Alcover_, Dec 07 2015 *)
%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^6)/n)}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=local(F=exp(sum(m=1, n, Lucas(m)^7*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. A203807, A203853, A203854, A203855, A203856, A203858, A203859, A203800.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 07 2012
| 