A209448 a(n) = Pell(n)*A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x)+theta_2(x)*theta_2(3*x))^4.

1, 24, 432, 4440, 21024, 87696, 559440, 1395264, 5728320, 23852760, 64719648, 183528288, 898460640, 1765134672, 6002425728, 21820957200, 52895150208, 134056553904, 598084104240, 1090757945760, 3530801969856, 11795485116480, 26821191064896, 65724336729792

Offset: 0

Author

Paul D. Hanna, Mar 10 2012

Keywords

• nonn

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)).

Examples

			G.f.: A(x) = 1 + 24*x + 432*x^2 + 4440*x^3 + 21024*x^4 + 87696*x^5 +...
where A(x) = 1 + 1*24*x + 2*216*x^2 + 5*888*x^3 + 12*1752*x^4 + 29*3024*x^5 +...+ Pell(n)*A008655(n)*x^n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A008655, A205968, A209447, A209449, A204270, A000129 (Pell), A002203.

Programs

Mathematica

A008655[n_]:= SeriesCoefficient[((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, n}]; b:= Table[A008655[n], {n, 0, 102}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n, 2]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 26 2018 *)

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 + sum(m=1,n, 24*Pell(m)*m^3*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*Pell(3*m)*(3*m)^3*x^(3*m)/(1-A002203(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n))  ),n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f.: 1 + Sum_{n>=1} 24*Pell(n)*n^3*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + 8*Pell(3*n)*(3*n)^3*x^(3*n)/(1 - A002203(3*n)*x^(3*n) + (-1)^n*x^(6*n)), where A002203(n) = Pell(n-1) + Pell(n+1).