login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Theta series of quadratic form with Gram matrix [ 2, 1; 1, 12 ].
11

%I #15 Mar 12 2021 22:24:41

%S 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,

%T 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,

%U 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

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

%C 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.

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

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

%D 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.

%H John Cannon, <a href="/A028959/b028959.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(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

%F 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

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

%e 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 + ...

%e 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 + ...

%t 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 *)

%Y Cf. A028958.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.