OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A008655:
1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 + Sum_{n>=1} 24*Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) + 8*Fibonacci(3*n)*(3*n)^3*x^(3*n)/(1 - Lucas(3*n)*x^(3*n) + (-1)^n*x^(6*n)).
EXAMPLE
G.f.: A(x) = 1 + 24*x + 216*x^2 + 1776*x^3 + 5256*x^4 + 15120*x^5 + ...
where A(x) = 1 + 1*24*x + 1*216*x^2 + 2*888*x^3 + 3*1752*x^4 + 5*3024*x^5 + ... + Fibonacci(n)*A008655(n)*x^n + ...
MATHEMATICA
A008655 := CoefficientList[Series[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, 250}], q]; b := Table[A008655[[n]], {n, 1, 120}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jul 16 2018 *)
PROG
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + sum(m=1, n, 24*fibonacci(m)*m^3*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*fibonacci(3*m)*(3*m)^3*x^(3*m)/(1-Lucas(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n)) ), n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2012
STATUS
approved