A033719 - OEIS

%I #26 Feb 04 2019 03:00:17

%S 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,

%T 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,

%U 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

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

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

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

%H Seiichi Manyama, <a href="/A033719/b033719.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%H G. E. Andrews, R. Lewis and Z.-G. Liu, <a href="http://dx.doi.org/10.1112/blms/33.1.25">An identity relating a theta series to a sum of Lambert series</a>, Bull. London Math. Soc., 33 (2001), 25-31.

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

%F 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, ...].

%F 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.

%F 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.

%e 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 + ...

%p 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

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

%o (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))};

%K nonn

%O 0,2

%A _N. J. A. Sloane_