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%I #28 Sep 30 2018 20:24:13
%S 1,112,1136,3136,9328,14112,31808,38528,74864,84784,143136,149184,
%T 261184,246176,390784,395136,599152,550368,859952,768320,1175328,
%U 1078784,1513152,1362816,2096192,1764112,2496928,2289280,3208832,2731680,4007808,3336704
%N Theta series of D_8 lattice.
%H Seiichi Manyama, <a href="/A008430/b008430.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from G. C. Greubel)
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 118.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%F G.f.: (theta_3(q^(1/2))^8 + theta_4(q^(1/2))^8)/2.
%F a(n) = A000143(2n).
%e 1 + 112*q^2 + 1136*q^4 + 3136*q^6 + 9328*q^8 + ...
%t 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 *)
%o (PARI) {a(n)=if(n<1, n==0, 16*sumdiv(n, d, d^3*(8-d%2)))} /* _Michael Somos_, Nov 03 2006 */
%o (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 */
%Y Cf. A000143, A008427 (dual), A109773.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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