A209454 a(n) = Pell(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).

1, 2, 0, 0, 24, 0, 0, 338, 1632, 1970, 0, 22964, 0, 0, 0, 0, 2824992, 0, 0, 0, 0, 0, 0, 900234724, 0, 2623476242, 0, 0, 36915112104, 178241928596, 0, 0, 5016108528384, 0, 0, 0, 42600007379160, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556, 98320743200877072, 0, 0, 0, 0

Offset: 0

Author

Paul D. Hanna, Mar 10 2012

Keywords

• nonn

Comments

Compare g.f. to 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n) (the Lambert series of A033719).

Examples

			G.f.: A(x) = 1 + 2*x + 24*x^4 + 338*x^7 + 1632*x^8 + 1970*x^9 + 22964*x^11 +...
where A(x) = 1 + 1*2*x + 12*2*x^4 + 169*2*x^7 + 408*4*x^8 + 985*2*x^9 + 5741*4*x^11 + 470832*6*x^16 + 225058681*4*x^23 +...+ Pell(n)*A033719(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1+2*x-x^2) + 2*x^2/(1-6*x^2+x^4) - 5*x^3/(1+14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
		

Links

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

Crossrefs

Cf. A033719, A205974, A209453, A209455, A204270, A000129 (Pell), A002203.

Programs

Mathematica

A033719[n_]:= SeriesCoefficient[EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^7], {x, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A033719[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)

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 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))

Formula

G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).