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%I #21 Sep 08 2022 08:44:48
%S 1,0,0,0,0,0,0,16,0,0,0,0,56,0,0,112,126,0,0,0,0,0,0,336,0,0,0,0,576,
%T 0,0,672,756,0,0,0,0,0,0,1232,0,0,0,0,1512,0,0,2016,2072,0,0,0,0,0,0,
%U 2800,0,0,0,0,4032,0,0,4048,4158,0,0,0,0,0,0,5712,0,0,0,0,5544,0,0,6944,7560
%N Theta series of A*_7 lattice. Expansion of F_8(q^2).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%H G. C. Greubel, <a href="/A023919/b023919.txt">Table of n, a(n) for n = 0..10000</a>
%H S. Ahlgren, <a href="http://dx.doi.org/10.1090/S0002-9939-99-05181-3">The sixth, eighth, ninth and tenth powers of Ramanujan's theta function</a>, Proc. Amer. Math. Soc., 128 (1999), 1333-1338.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As7.html">Home page for this lattice</a>
%e G.f. = 1 + 16*q^7 + 56*q^12 + 112*q^15 + 126*q^16 + 336*q^23 + 576*q^28 + 672*q^31 + 756*q^32 + 1232*q^39 + 1512*q^44 + 2016*q^47 + 2072*q^48 + O(q^49)
%t terms = 81; phi[q_] := EllipticTheta[3, 0, q]; psi[q_] := (1/2)*q^(-1/8) * EllipticTheta[2, 0, q^(1/2)]; F8[q_] := (1/8) (phi[q^2]^7 + (2 Sqrt[q] psi[q^4])^7 + 14 phi[q^2]^5 phi[q]^2 - 7 phi[q^2]^3 phi[q]^4); s = Simplify[Normal[F8[q^2] + O[q]^terms], q>0]; CoefficientList[s, q][[1 ;; terms]] (* _Jean-François Alcover_, Jul 04 2017 *)
%o (Magma) L:=Lattice("A",7); D:=Dual(L); T1<q> := ThetaSeries(D,60);
%Y Cf. A008447 (A_7).
%K nonn
%O 0,8
%A _N. J. A. Sloane_
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