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A023921 Theta series of A*_9 lattice.

%I A023921 #13 Jul 08 2025 17:22:28
%S A023921 1,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,90,0,0,0,90,240,0,0,420,252,0,0,0,
%T A023921 720,0,0,0,0,0,0,1700,0,0,0,1260,2700,0,0,3240,1680,0,0,0,5220,0,0,0,
%U A023921 0,0,0,7740,0,0,0,4920,10080,0,0,12690,7020,0,0,0,15120,0,0,0,0,0
%N A023921 Theta series of A*_9 lattice.
%D A023921 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.
%H A023921 G. C. Greubel, <a href="/A023921/b023921.txt">Table of n, a(n) for n = 0..5000</a>
%H A023921 S. Ahlgren, <a href="https://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. [Expansion of F_10(q^2)]
%H A023921 G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/As9.html">Home page for this lattice</a>
%e A023921 G.f. = 1 + 20*q^9 + 90*q^16 + 90*q^20 + 240*q^21 + 420*q^24 + 252*q^25 + 720*q^29 + 1700*q^36 + 1260*q^40 + .... - _G. C. Greubel_, Apr 16 2018
%t A023921 f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; F[10,q_]:= f[q^5, q^5]^10/f[q, q] - 8*q^2*f[q^5, q^5]^5* QPochhammer[q^10]^5/QPochhammer[q^2] - (27*q^4*f[q^5, q^5]^5 - 125*q^9*f[q^5, q^5]*f[q^10, q^30]^4 + 5*q^5*f[q^5, q^5]*f[q^2, q^6]^4)* QPochhammer[q^20]^5/QPochhammer[q^4] - 17*q^5*f[q, q]*f[q^5, q^15]^8 + 2*q*f[q, q]*f[q^5, q^5]^4*f[q^2, q^6]^4 - 20*q^8*f[q, q]*QPochhammer[q^20]^10/QPochhammer[q^4]^2 + 5*q^4*f[q, q]*QPochhammer[q^10]^10/QPochhammer[q^2]^2; CoefficientList[Series[F[10, q], {q, 0, 100}], q] (* _G. C. Greubel_, Apr 16 2018 *)
%Y A023921 Cf. A072836.
%K A023921 nonn
%O A023921 0,10
%A A023921 _N. J. A. Sloane_