A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160

Offset

0,2

Comments

Compare g.f. to the Lambert series of A113973:

1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

Links

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

Formula

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

Example

G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(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) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

Mathematica

A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)

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 - 2*sum(m=1, n, Pell(m)*kronecker(m, 3)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Cf. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.

Sequence in context: A122208 A297475 A106358 * A002510 A247592 A102943

Adjacent sequences: A209446 A209447 A209448 * A209450 A209451 A209452

Keyword

sign

Author

Paul D. Hanna, Mar 10 2012

Status

approved