|
%I #34 Oct 02 2018 09:27:43
%S 1,0,4320,61440,522720,2211840,8960640,23224320,67154400,135168000,
%T 319809600,550195200,1147643520,1771683840,3371915520,4826603520,
%U 8593797600,11585617920,19590534240,25239859200,40979580480,50877235200
%N Theta series of 16-dimensional Barnes-Wall lattice.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 130, p. 131 Equation (132).
%H Seiichi Manyama, <a href="/A008409/b008409.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/BW16.html">Home page for this lattice</a>
%H <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ThetaSeries.html">Theta Series</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Barnes-WallLattice.html">Barnes-Wall Lattice</a>
%F Expansion of ( theta_2(q)^16 + theta_3(q)^16 + theta_4(q)^16 + 30 * theta_2(q)^8 * theta_3(q)^8 ) / 2 in powers of q. - [Conway and Sloane]
%F Expansion of E_4(q^2)^2 + (E_4(q) - E_4(q^2))^2 / 15 in powers of q. - _Michael Somos_, Nov 29 2007
%F Expansion of ( eta(q)^48 + 32 * eta(q)^24 * eta(q^2)^24 + 4096 * eta(q^2)^48 ) / ( eta(q) * eta(q^2) )^16 in powers of q. - _Michael Somos_, Nov 29 2007
%F G.f. is Fourier series of a weight 8 level 2 modular form. f(-1 / (2 t)) = 16 (t/i)^8 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Nov 29 2007
%e 1 + 4320*q^4 + 61440*q^6 + 522720*q^8 + 2211840*q^10 + 8960640*q^12 + ...
%t f[q_] := 1/2*(EllipticTheta[2, 0, q]^16 + EllipticTheta[3, 0, q]^16 + EllipticTheta[4, 0, q]^16 + 30*EllipticTheta[2, 0, q]^8*EllipticTheta[3, 0, q]^8); Series[f[q], {q, 0, 21}] // CoefficientList[#, q]& (* _Jean-François Alcover_, May 15 2013 *)
%o (PARI) {a(n) = local(A1, A2) ; if( n<0, 0, A1 = eta(x + x * O(x^n))^8; A2 = eta(x^2 + x * O(x^n))^8; polcoeff( (A1^6 + 32 * x * A1^3 * A2^3 + 4096 * x^2 * A2^6) / ( A1 * A2 )^2, n))} /* _Michael Somos_, Nov 29 2007 */
%Y A008774(2*n) = a(n).
%Y Cf. A004009, A319307.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
|
