A028956 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 4 ].

1, 0, 4, 2, 0, 2, 0, 0, 8, 0, 0, 0, 6, 0, 0, 0, 0, 4, 4, 0, 6, 0, 0, 4, 0, 0, 0, 2, 0, 0, 4, 0, 12, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 2, 0, 4, 10, 0, 4, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 12, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 4, 0, 4, 0, 0, 4, 0, 0, 0, 8

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The number of integer solutions (x, y) to 2*x^2 + x*y + 2*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014

Examples

			G.f. =  1 + 4*q^4 + 2*q^6 + 2*q^10 + 8*q^16 + 6*q^24 + 4*q^34 + 4*q^36 + 6*q^40 + 4*q^46 + 2*q^54 + 4*q^60 + 12*q^64 + 8*q^76 + 2*q^90 + 4*q^94 + 10*q^96 + 4*q^100 + ...
G.f. = 1 + 4*x^2  + 2*x^3 + 2*x^5  + 8*x^8 + 6*x^12 + 4*x^17 + ... - _Michael Somos_, Jan 23 2023
		

References

• R. Barman and N. D. Baruah, Theta function identities associated with Ramanujan's modular equations of degree 15, Proc. Indian Acad. Sci. Math. Sci. 120 (2010), no. 3, 267-284. see p. 271, equ. (3.1)

Links

• John Cannon, Table of n, a(n) for n = 0..5000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A192323.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^5], {q, 0, 4 n}]; (* Michael Somos, Aug 01 2011 *)

PARI

{a(n) = if( n<1, n==0, qfrep([4, 1;1, 4], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */

PARI

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d) - kronecker( -3, d) * kronecker( 5, n/d)))}; /* Michael Somos, Aug 26 2006 */

Formula

Expansion of phi(q^3) * phi(q^5) + 4 * q^2 * psi(q^6) * psi(q^10) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, Feb 09 2006

Expansion of (phi(q^3) * phi(q^5) + phi(-q^3) * phi(-q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function. - Michael Somos, Aug 01 2011

Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) - (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q. - Michael Somos, Aug 26 2006

G.f.: theta_3(q^3) * theta_3(q^5) + theta_2(q^3) * theta_2(q^5) . - Michael Somos, Feb 09 2006

a(n) = A192323(4*n).