A028959 Theta series of quadratic form with Gram matrix [ 2, 1; 1, 12 ].

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

Offset

0,2

Comments

theta[2,1;1,2d](z)=theta_3(z)*theta_3((4d-1)z)+theta_2(z)*theta_2((4d-1)z), generalizing the formula for theta(A_2), which is the case d=1 - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 16 2000.

The number of integer solutions (x, y) to x^2 + x*y + 6*y^2 = n, discriminant -23. - Ray Chandler, Jul 12 2014

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

References

Köklüce, Bülent. "Cusp forms in S_6 (Gamma_ 0(23)), S_8 (Gamma_0 (23)) and the number of representations of numbers by some quadratic forms in 12 and 16 variables." The Ramanujan Journal 34.2 (2014): 187-208. See F_1, p. 195.

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(x) * phi(x^23) + 4*x^6 * psi(x^2) * psi(x^46) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 28 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 28 2015

G.f.: (theta_3(z)*theta_3(23z) + theta_2(z)*theta_2(23z)).

Example

G.f. = 1 + 2*x + 2*x^4 + 4*x^6 + 4*x^8 + 2*x^9 + 4*x^12 + 2*x^16 + 4*x^18 + ...
G.f. = 1 + 2*q^2 + 2*q^8 + 4*q^12 + 4*q^16 + 2*q^18 + 4*q^24 + 2*q^32 + 4*q^36 + 2*q^46 + 4*q^48 + 2*q^50 + 4*q^52 + 4*q^54 + 4*q^64 + 6*q^72 + 4*q^78 + 8*q^96 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^23] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^23], {x, 0, n}]; (* Michael Somos, Mar 28 2015 *)

Crossrefs

Cf. A028958.

Sequence in context: A263146 A365047 A361015 * A317642 A258762 A079181

Adjacent sequences: A028956 A028957 A028958 * A028960 A028961 A028962

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved