A209450 a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

1, -3, 6, -15, 36, 0, 210, -1014, 1224, -2955, 0, 0, 41580, -200766, 484692, 0, 1412496, 0, 8232630, -39750654, 0, -231683790, 0, 0, 1630019160, -3935214363, 19000895772, -22936110135, 110745336312, 0, 0, -1558305137094, 1881040698144, 0, 0, 0, 63900011068740

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

G.f.: 1 - 3*Sum_{n>=0} Pell(6*n+1)*x^(6*n+1)/(1+A002203(6*n+1)*x^(6*n+1)-x^(12*n+2)) - Pell(6*n+5)*x^(6*n+5)/(1+A002203(6*n+5)*x^(6*n+5)-x^(12*n+10)), where A002203(n) = Pell(n-1) + Pell(n+1).

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

Cf. A132973, A205970, A209449, A209451, A204270, A000129 (Pell), A002203.

Sequence in context: A190586 A113225 A298539 * A291013 A017924 A052827

Adjacent sequences: A209447 A209448 A209449 * A209451 A209452 A209453

Keyword

sign

Author

Paul D. Hanna, Mar 10 2012

Status

approved