A209456 a(n) = Pell(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.

1, 4, 24, 80, 336, 696, 3360, 676, 24480, 51220, 171216, 275568, 1552320, 1873816, 969384, 18722400, 58383168, 81841608, 428096760, 530008720, 2687063904, 617823440, 13424019552, 21605633376, 130401532800, 162655527004, 532025081616, 1223259207200

Offset: 0

Author

Paul D. Hanna, Mar 10 2012

Keywords

• nonn

Comments

Compare g.f. to the Lambert series of A028594:

1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).

Here Chi(n,7) = principal Dirichlet character of n modulo 7.

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 80*x^3 + 336*x^4 + 696*x^5 + 3360*x^6 +...
where A(x) = 1 + 1*4*x + 2*12*x^2 + 5*16*x^3 + 12*28*x^4 + 29*24*x^5 + 70*48*x^6 + 169*4*x^7 + 408*60*x^8 + 985*52*x^9 +...+ Pell(n)*A028594(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1-198*x^6+x^12) + 0*169*7*x^7/(1+478*x^7-x^14) +...).
The values of the Dirichlet character Chi(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. A028594, A205976, A209455, A204270, A000129 (Pell), A002203.

Programs

Mathematica

A028594[n_]:= If[n < 1, Boole[n == 0], 4*Sum[If[Mod[d, 7] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A028594[n], {n,1,50}]] (* G. C. Greubel, Jan 03 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 + 4*sum(m=1,n,Pell(m)*kronecker(m,7)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f.: 1 + 4*Sum_{n>=1} Pell(n)*Chi(n,7)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).