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a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.
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%I #9 Aug 22 2023 07:59:38

%S 4,16,64,280,1344,6496,32640,166320,862400,4523232,23970240,128063040,

%T 689008320,3728973120,20285199872,110841302880,608029121280,

%U 3346972244000,18480871268160,102328688556864,568014587806720,3160148362953120,17617881702072960

%N a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.

%F G.f.: 1/Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} A002203(n)^3 * x^n/n), which equals the g.f. of A212442.

%t a[n_] := DivisorSum[n, MoebiusMu[n/#] * LucasL[#, 2]^2 &] / n; Array[a, 25] (* _Amiram Eldar_, Aug 22 2023 *)

%o (PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

%o {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*A002203(d)^2)/n)}

%o for(n=1,30,print1(a(n),","))

%Y Cf. A008683, A212442, A203853, A002203.

%K nonn

%O 1,1

%A _Paul D. Hanna_, May 17 2012