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Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 4 ].
2

%I #24 Jan 25 2023 10:03:21

%S 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,

%T 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,

%U 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

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

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

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

%D 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)

%H John Cannon, <a href="/A028956/b028956.txt">Table of n, a(n) for n = 0..5000</a>

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F 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

%F 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

%F G.f.: theta_3(q^3) * theta_3(q^5) + theta_2(q^3) * theta_2(q^5) . - _Michael Somos_, Feb 09 2006

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

%e 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 + ...

%e 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

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

%o (PARI) {a(n) = if( n<1, n==0, qfrep([4, 1;1, 4], n, 1)[n]*2)}; /* _Michael Somos_, Aug 26 2006 */

%o (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 */

%Y Cf. A192323.

%K nonn

%O 0,3

%A _N. J. A. Sloane_.