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

0,3

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

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

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).

Example

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

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A192323.

Sequence in context: A049430 A055356 A297331 * A129681 A126079 A106220

Adjacent sequences: A028953 A028954 A028955 * A028957 A028958 A028959

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved