A008430 Theta series of D_8 lattice.

1, 112, 1136, 3136, 9328, 14112, 31808, 38528, 74864, 84784, 143136, 149184, 261184, 246176, 390784, 395136, 599152, 550368, 859952, 768320, 1175328, 1078784, 1513152, 1362816, 2096192, 1764112, 2496928, 2289280, 3208832, 2731680, 4007808, 3336704

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			1 + 112*q^2 + 1136*q^4 + 3136*q^6 + 9328*q^8 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from G. C. Greubel)
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Crossrefs

Cf. A000143, A008427 (dual), A109773.

Programs

Mathematica

a[n_] := 16*DivisorSum[n, #^3*(8 - Mod[#, 2]) &]; a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

PARI

{a(n)=if(n<1, n==0, 16*sumdiv(n, d, d^3*(8-d%2)))} /* Michael Somos, Nov 03 2006 */

PARI

{a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^8, n))} /* Michael Somos, Nov 03 2006 */

Formula

G.f.: (theta_3(q^(1/2))^8 + theta_4(q^(1/2))^8)/2.

a(n) = A000143(2n).