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%I #16 Dec 01 2017 18:59:33
%S 1,1,5,85,2928,314925,84974760,63327890015,123670531939440,
%T 644385861467631972,8853970669063185618000,
%U 321538767413685546538468385,30768712746239178236068160093280,7755868453482819803691622493685140880,5144106193113274410507722020733942141881664
%N a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).
%H G. C. Greubel, <a href="/A203800/b203800.txt">Table of n, a(n) for n = 1..69</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)^n * x^n/n), which is the g.f. of A156216.
%F G.f.: Product_{n>=1} G_n(x^n)^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.
%e G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^85 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^314925 * (1-29*x^7-x^14)^84974760 * (1-47*x^8+x^16)^63327890015 * (1-76*x^9-x^18)^123670531939440 *...).
%e where F(x) = exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = g.f. of A156216:
%e F(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
%e so that the logarithm of F(x) begins:
%e log(F(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 + 29^7*x^7/7 + 47^8*x^8/8 + 76^9*x^9/9 + 123^10*x^10/10 +...+ Lucas(n)^n*x^n +...
%t a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^(#-1)&]; Array[a, 15] (* _Jean-François Alcover_, Dec 23 2015 *)
%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^(d-1))/n)}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=local(F=exp(sum(m=1, n, Lucas(m)^m*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. A156216, A000032 (Lucas), A203318, A203803, A203804, A203805, A203806, A203807, A203808.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jan 06 2012
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