%I #35 Jan 21 2024 02:20:12

%S 1,12,36,12,84,72,36,96,180,12,216,144,84,168,288,72,372,216,36,240,

%T 504,96,432,288,180,372,504,12,672,360,216,384,756,144,648,576,84,456,

%U 720,168,1080,504,288,528,1008,72,864,576,372,684,1116,216,1176,648,36

%N Theta series of direct sum of 2 copies of hexagonal lattice.

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

%C Convolution square of A004016.

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

%C Denoted by E_{2,3}^{i\infinity}(\tau) in Kaneko and Sakai 2012 on page 7. - _Michael Somos_, Dec 27 2014

%D Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 460, Entry 3(i).

%D J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 110.

%H Seiichi Manyama, <a href="/A008653/b008653.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>.

%H Masanobu Kaneko and Yuichi Sakai, <a href="http://arxiv.org/abs/1201.1685">The Ramanujan-Serre Differential Operators and certain Elliptic Curves</a>, arXiv:1201.1685 [math.NT], 2012.

%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

%H Gabriele 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>.

%F Expansion of (theta_3(z)*theta_3(3z)+theta_2(z)*theta_2(3z))^2.

%F Expansion of a(q)^2 in powers of q where a() is a cubic AGM theta function.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 9*v^2 + 16*w^2 - 6*u*v + 4*u*w - 24*v*w. - _Michael Somos_, Jul 19 2004

%F G.f.: 1 + 12* Sum_{k>0} x^k / (1 - x^k)^2 - 36* Sum_{k>0} x^(3*k) / (1 - x^(3*k))^2. - _Michael Somos_, Apr 15 2007

%F a(n) = 12 * A046913(n) unless n=0.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/3. - _Amiram Eldar_, Jan 21 2024

%e G.f. = 1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + ...

%t a[ n_] := SeriesCoefficient[ ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* _Michael Somos_, May 26 2014 *)

%t a[ n_] := If[ n < 1, Boole[ n == 0], 12 Sum[ If[ Mod[ d, 3] > 0, d, 0], {d, Divisors @ n }]]; (* _Michael Somos_, May 26 2014 *)

%o (PARI) {a(n) = if( n<1, n==0, 12 * (sigma(3*n) - 3*sigma(n)))}; /* _Michael Somos_, Jul 19 2004 */

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, 6 * x^k / (1 + x^k + x^(2*k)), 1 + x * O(x^n))^2, n))}; /* _Michael Somos_, Jul 19 2004 */

%o (Sage) ModularForms( Gamma0(3), 2, prec=70).0; # _Michael Somos_, Jun 12 2014

%o (Magma) Basis( ModularForms( Gamma0(3), 2), 70)[1]; /* _Michael Somos_, Jun 12 2014 */

%Y Cf. A004016, A046913, A005882, A005928.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_