A209614 - OEIS

%I #6 Apr 07 2012 22:51:00

%S 1,0,0,0,0,0,0,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,196418,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

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

%N G.f.: Sum_{n>=1} Fibonacci(n^3)*x^(n^3).

%C Compare g.f. to the Lambert series identity: Sum_{n>=1} A210826(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^3), where A210826(n) = kronecker(-3,d(n)) and d(n) is the number of divisors of n.

%F G.f.: Sum_{n>=1} A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} Fibonacci(n^3)*x^(n^3).

%e G.f.: A(x) = x + 21*x^8 + 196418*x^27 + 10610209857723*x^64 + 59425114757512643212875125*x^125 +...

%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) + (0)*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)+ (-1)*13*x^7/(1-29*x^7-x^14) + (+1)*21*x^8/(1-47*x^8+x^16) + (0)*34*x^9/(1-76*x^9-x^18) + (+1)*55*x^10/(1-123*x^10+x^20)+ (-1)*89*x^11/(1-199*x^11-x^22) +...+ A210826(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

%e The sequence A210826(n) = kronecker(-3, d(n)) begins:

%e [1,-1,-1,0,-1,1,-1,1,0,1,-1,0,-1,1,1,-1,-1,0,-1,0,1,1,-1,-1,0,1,1,...].

%o (PARI) {A210826(n)=if(n==0,0,kronecker(-3,numdiv(n)))}

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

%o {a(n)=polcoeff(sum(m=1, n, A210826(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}

%o for(n=1,125,print1(a(n),", "))

%Y Cf. A210826, A204060.

%K nonn

%O 1,8

%A _Paul D. Hanna_, Apr 07 2012