OFFSET
1,4
COMMENTS
Compare g.f. to the Lambert series identity: Sum_{n>=1} lamda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2).
Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).
FORMULA
EXAMPLE
G.f.: A(x) = x + 3*x^4 + 34*x^9 + 987*x^16 + 75025*x^25 + 14930352*x^36 +...
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)) +...
PROG
(PARI) {a(n)=issquare(n)*fibonacci(n)}
(PARI) {lambda(n)=local(F=factor(n)); (-1)^sum(i=1, matsize(F)[1], F[i, 2])}
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 12 2012
STATUS
approved