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%I #18 Sep 08 2022 08:44:48
%S 1,0,0,0,0,12,0,0,30,20,0,0,30,0,0,0,0,120,0,0,132,60,0,0,90,0,0,0,0,
%T 180,0,0,270,180,0,0,140,0,0,0,0,480,0,0,420,132,0,0,270,0,0,0,0,420,
%U 0,0,600,420,0,0,360,0,0,0,0,960,0,0,840,360,0,0,330,0,0,0,0
%N Theta series of A*_5 lattice.
%C Expansion of Ahlgren's F_6(q^2).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%H John Cannon, <a href="/A023917/b023917.txt">Table of n, a(n) for n = 0..10000</a>
%H S. Ahlgren, <a href="https://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 (2000), 1333-1338.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As5.html">Home page for this lattice</a>
%e 1 + 12*q^5 + 30*q^8 + 20*q^9 + 30*q^12 + 120*q^17 + 132*q^20 + 60*q^21 + 90*q^24 + 180*q^29 + 270*q^32 + 180*q^33 + 140*q^36 + 480*q^41 + 420*q^44 + 132*q^45 + 270*q^48 + 420*q^53 + 600*q^56 + 420*q^57 + 360*q^60 + O(q^61)
%t terms = 77; phi[q_] := EllipticTheta[3, 0, q]; F6[q_] := (1/32)*(-3*phi[Sqrt[q]]^5 + 5*phi[Sqrt[q]]^3*phi[Sqrt[q^3]]^2 + 15*phi[Sqrt[q]] * phi[Sqrt[q^3]]^4 + (15*phi[Sqrt[q^3]]^6)/phi[Sqrt[q]]); s = Simplify[F6[q^2], q>0]; s = s + O[q]^(2 terms); CoefficientList[s, q][[1 ;; terms]] (* _Jean-François Alcover_, Jul 04 2017 *)
%o (Magma) L:=Lattice("A",5); D:=Dual(L); T1<q> := ThetaSeries(D,120);
%K nonn
%O 0,6
%A _N. J. A. Sloane_.
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