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A033728 Product theta3(q^d); d | 16.

%I A033728 #13 Oct 06 2018 11:44:35
%S A033728 1,2,2,4,4,4,8,8,8,10,12,12,16,20,16,24,26,24,30,28,32,40,40,40,48,50,
%T A033728 52,64,64,60,80,80,72,88,88,88,100,100,88,104,112,112,120,124,112,124,
%U A033728 144,112,144,146,134,168,160,140,160,184,160,184,188,148,192,220,160
%N A033728 Product theta3(q^d); d | 16.
%D A033728 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
%H A033728 Seiichi Manyama, <a href="/A033728/b033728.txt">Table of n, a(n) for n = 0..10000</a>
%F A033728 Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8) * eta(q^16) * eta(q^32)^3 / (eta(q)^2 * eta(q^64)^2) in powers of q.
%e A033728 G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 8*x^8 + ...
%t A033728 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^2] EllipticTheta[ 3, 0, x^4] EllipticTheta[ 3, 0, x^8] EllipticTheta[ 3, 0, x^16], {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%o A033728 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=0, 4, sum(i=1, sqrtint( n\(2^k)), 2 * x^(2^k * i^2), 1 + x*O(x^n))), n))};
%o A033728 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(X)^-2 * eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A) * eta(x^16 + A) * eta(x^32 + A)^3 * eta(x^64 + A)^-2, n))};
%K A033728 nonn
%O A033728 0,2
%A A033728 _N. J. A. Sloane_