%I #12 Dec 31 2012 03:05:02

%S 1,0,0,3,0,0,0,0,34,0,0,0,0,0,0,987,0,0,0,0,0,0,0,0,75025,0,0,0,0,0,0,

%T 0,0,0,0,14930352,0,0,0,0,0,0,0,0,0,0,0,0,7778742049,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,10610209857723,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,37889062373143906

%N G.f.: Sum_{n>=1} Fibonacci(n^2)*x^(n^2).

%C Compare g.f. to the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2).

%C Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

%F G.f.: Sum_{n>=1} lambda(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)), where lambda(n) = A008836(n) and Lucas(n) = A000204(n).

%e G.f.: A(x) = x + 3*x^4 + 34*x^9 + 987*x^16 + 75025*x^25 + 14930352*x^36 +...

%e where A(x) = x/(1-x-x^2) + (-1)*1*x^2/(1-3*x^2+x^4) + (-1)*2*x^3/(1-4*x^3-x^6) + (+1)*3*x^4/(1-7*x^4+x^8) + (-1)*5*x^5/(1-11*x^5-x^10) + (+1)*8*x^6/(1-18*x^6+x^12) +...+ lambda(n)*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

%o (PARI) {a(n)=issquare(n)*fibonacci(n)}

%o (PARI) {lambda(n)=local(F=factor(n));(-1)^sum(i=1,matsize(F)[1],F[i,2])}

%o {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

%o {a(n)=polcoeff(sum(m=1,n,lambda(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

%Y Cf. A203847, A054783, A008836 (lambda), A000204 (Lucas), A000045.

%Y Cf. A209614 (variant).

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jan 12 2012