OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1 + 16*Sum_{n>=1} Fibonacci(n)*n^3*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
EXAMPLE
G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
where A(x) = 1 + 1*16*x + 1*112*x^2 + 2*448*x^3 + 3*1136*x^4 + 5*2016*x^5 + 8*3136*x^6 + 13*5504*x^7 + 21*9328*x^8 +...+ Fibonacci(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+x-x^2) + 1*8*x^2/(1-3*x^2+x^4) + 2*27*x^3/(1+4*x^3-x^6) + 3*64*x^4/(1-7*x^4+x^8) + 5*125*x^5/(1+11*x^5-x^10) + 8*216*x^6/(1-18*x^6+x^12) + 13*343*x^7/(1+29*x^7-x^14) +...).
MATHEMATICA
Join[{1}, Table[Fibonacci[n]*SquaresR[8, n], {n, 1, 50}]] (* G. C. Greubel, Mar 05 201 *)
PROG
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+16*sum(m=1, n, fibonacci(m)*m^3*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 03 2012
STATUS
approved