%I #29 Sep 08 2022 08:44:36

%S 1,18,108,234,234,864,756,900,1836,2178,1296,4320,3042,3060,5400,6048,

%T 3690,10368,6588,6516,11232,11700,6480,19008,12852,10818,18360,19674,

%U 11700,30240,16848,17316,29484,30240,15552,43200,28314,24660,39096

%N Theta series of direct sum of 3 copies of hexagonal lattice.

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

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

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

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 110.

%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 124, Equation (7.19).

%H Seiichi Manyama, <a href="/A008654/b008654.txt">Table of n, a(n) for n = 0..10000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>

%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 (theta_3(z)*theta_3(3z) + theta_2(z)*theta_2(3z))^3.

%F Expansion of a(q)^3 in powers of q where a() is a cubic AGM function. - _Michael Somos_, Sep 04 2008

%F 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

%F 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

%F 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

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

%t 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 *)

%o (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 */

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Michael Somos_, Sep 04 2008