A033719 Coefficients in expansion of theta_3(q) * theta_3(q^7) in powers of q.

1, 2, 0, 0, 2, 0, 0, 2, 4, 2, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 4, 0, 0, 8, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 4, 4, 0, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of integer solutions to the equation x^2 + 7*y^2 = n.

Examples

			G.f. = 1 + 2*x + 2*x^4 + 2*x67 + 4*x^8 + 2*x^9 + 4*x^11 + 6*x^16 + 4*x^23 + ...
		

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 (terms 0..1000 from G. C. Greubel)
• G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.

Programs

Maple

seq(coeff(series(mul((1-x^(2*k))*(1+x^(2*k-1))^2*(1-x^(14*k))*(1+x^(14*k-7))^2,k=1..n),x,n+1), x, n), n = 0 .. 110); # Muniru A Asiru, Feb 02 2019

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^7], {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2, n))};

Formula

Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2 + 7*j^2).

Euler transform of period 28 sequence [ 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -2, ...].

Expansion of (eta(q^2) * eta(q^14))^5 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28))^2 in powers of q.

G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^2 * (1 - x^(14*k)) * (1 + x^(14*k-7))^2.