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

0,2

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.

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

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

Example

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

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 *)

Prog

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

Crossrefs

Sequence in context: A317461 A225219 A002165 * A060787 A019584 A356420

Adjacent sequences: A008651 A008652 A008653 * A008655 A008656 A008657

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Sep 04 2008

Status

approved