A008654 Theta series of direct sum of 3 copies of hexagonal lattice.

1, 18, 108, 234, 234, 864, 756, 900, 1836, 2178, 1296, 4320, 3042, 3060, 5400, 6048, 3690, 10368, 6588, 6516, 11232, 11700, 6480, 19008, 12852, 10818, 18360, 19674, 11700, 30240, 16848, 17316, 29484, 30240, 15552, 43200, 28314, 24660, 39096

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Examples

			G.f. = 1 + 18*q + 108*q^2 + 234*q^3 + 234*q^4 + 864*q^5 + 756*q^6 + 900*q^7 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.
• B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 124, Equation (7.19).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Magma

A := Basis( ModularForms( Gamma1(3), 3), 39); A[1] + 18*A[2]; /* Michael Somos, Aug 26 2015 */

Mathematica

a[ n_] := With[ {A = QPochhammer[ q]^3, A3 = QPochhammer[ q^3]^3}, SeriesCoefficient[ (A^4 + 27 q A3^4) / (A A3), {q, 0, n}]]; (* Michael Somos, Oct 22 2017 *)

PARI

{a(n) = my(A, A3); if( n<0, 0, A = x * O(x^n); A3 = eta(x^3 + A)^3; A = eta(x + A)^3; polcoeff( (A^4 + 27 * x * A3^4) / (A * A3), n))}; /* Michael Somos, Sep 04 2008 */

Formula

Expansion of (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^3.

Expansion of a(q)^3 in powers of q where a() is a cubic AGM function. - Michael Somos, Sep 04 2008

Expansion of (eta(q)^12 + 27 * eta(q^3)^12) / (eta(q) * eta(q^3))^3 in powers of q. - Michael Somos, Sep 04 2008

Expansion of (f(-q)^12 + 27 * q * f(-q^3)^12) / (f(-q) * f(-q^3))^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Sep 04 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 04 2008

Extensions

More terms from Michael Somos, Sep 04 2008