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

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, 52, 64, 64, 60, 80, 80, 72, 88, 88, 88, 100, 100, 88, 104, 112, 112, 120, 124, 112, 124, 144, 112, 144, 146, 134, 168, 160, 140, 160, 184, 160, 184, 188, 148, 192, 220, 160

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			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 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Programs

Mathematica

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 *)

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))};

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))};

Formula

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.