A028953 Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].

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

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 3*x^2 + 2*x*y + 4*y^2, discriminant -44. - Ray Chandler, Jul 12 2014

Examples

			G.f. = 1 + 2*q^3 + 2*q^4 + 2*q^5 + 2*q^9 + 4*q^12 + 2*q^15 + 2*q^16 + 4*q^20 + 2*q^23 + 2*q^25 + 2*q^27 + 2*q^31 + 2*q^33 + 6*q^36 + 2*q^37 + 2*q^44 + 4*q^45 + ...
		

Links

• John Cannon, Table of n, a(n) for n = 0..10000
• 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. A028952, A106282.

Programs

Magma

A := Basis( ModularForms( Gamma1(44), 1), 87); A[1] + 2*A[4] + 2*A[5] + 2*A[6] + 2*A[10] + 4*A[13] + 2*A[16] + 2*A[17] + 4*A[21] + 2*A[24]; /* Michael Somos, Feb 09 2017 */

Mathematica

r[n_] := Reduce[{x, y}.{{3, 1}, {1, 4}}.{x, y} == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] - 2 q QPochhammer[ q^2] QPochhammer[ q^22], {q, 0, n}]; (* Michael Somos, Feb 09 2017 *)

PARI

{a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)}; /* Michael Somos, Jun 24 2011 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\11), 2 * x^(11*k^2), 1 + A) - 2 * x * eta(x^2 + A) * eta(x^22 + A), n))}; /* Michael Somos, Jun 24 2011 */

Formula

Expansion of phi(q) * phi(q^11) - 2*q * f(-q^2) * f(-q^22) = phi(q^3) * phi(q^33) + 2*q^4 * chi(q) * psi(-q^3) * chi(q^11) * psi(-q^33) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos and Alex Berkovich, Jun 24 2011

G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 24 2011

G.f.: Sum_{n, m in Z} x ^ (3*n*n + 2*n*m + 4*m*m).

a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0. - Michael Somos, Feb 23 2012