OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A132973: 1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
EXAMPLE
G.f.: A(x) = 1 - 3*x + 6*x^2 - 15*x^3 + 36*x^4 + 210*x^6 - 1014*x^7 +...
where A(x) = 1 - 1*3*x + 2*3*x^2 - 5*3*x^3 + 12*3*x^4 + 70*3*x^6 - 169*6*x^7 + 408*3*x^8 +...+ Pell(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+2*x-x^2) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 5741*x^11/(1+16238*x^11-x^22) + 33461*x^13/(1+94642*x^13-x^26) - 1136689*x^17/(1+3215042*x^17-x^34) +...).
MATHEMATICA
A132973[n_]:= SeriesCoefficient[EllipticTheta[2, Pi/4, q^(1/2)]^3/EllipticTheta[2, Pi/4, q^(3/2)]/2, {q, 0, n}]; Join[{1}, Table[ Fibonacci[n, 2]*A132973[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)
PROG
(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 - 3*sum(m=0, n, Pell(6*m+1)*x^(6*m+1)/(1+A002203(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - Pell(6*m+5)*x^(6*m+5)/(1+A002203(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ), n)}
for(n=0, 61, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 10 2012
STATUS
approved