A046113 Coefficients in expansion of theta_3(q) * theta_3(q^6) in powers of q.

1, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 2, 6, 0, 0, 4, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 8, 0, 0, 4, 0, 4, 0, 0, 4, 2, 0, 0, 0, 0, 0, 8, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, 0

Offset

0,2

Comments

Number of representations of n as a sum of six times a square and a square. - Ralf Stephan, May 14 2007

a(n) < 2 if and only if n is in A002480. a(n) > 0 if and only if n is in A002481. - Michael Somos, Mar 01 2011

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms

Formula

G.f.: Sum_{ i, j = -inf .. inf } q^(i^2 + 6*j^2).

a(n) = A000377(n) + A115660(n). - Michael Somos, Mar 01 2011

Example

G.f. = 1 + 2*x + 2*x^4 + 2*x^6 + 4*x^7 + 2*x^9 + 4*x^10 + 4*x^15 + 2*x^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6], {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)

Prog

(PARI) {a(n) = my(G); if( n<0, 0, G = [ 1, 0; 0, 6]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Mar 01 2011 */

Crossrefs

Cf. A000377, A002480, A002481, A115660.

Sequence in context: A291289 A095759 A260309 * A262938 A143068 A261202

Adjacent sequences: A046110 A046111 A046112 * A046114 A046115 A046116

Keyword

nonn

Author

N. J. A. Sloane, May 18 2002

Status

approved