A317642 - OEIS

%I #4 Aug 03 2018 08:18:23

%S 1,0,2,0,0,2,0,4,2,0,0,0,0,4,0,0,0,0,2,0,2,0,4,4,0,0,0,0,4,0,0,0,2,0,

%T 0,0,0,4,4,0,0,0,0,0,0,2,0,4,0,0,2,0,4,4,0,4,0,0,0,0,0,0,0,4,0,0,0,0,

%U 0,0,4,0,2,0,0,0,0,8,0,0,2,0,4,0,0,0,0,0,4,0,0,0,4,0,0,4,0,0,6

%N Expansion of theta_3(q^2)*theta_3(q^5), where theta_3() is the Jacobi theta function.

%C Number of integer solutions to the equation 2*x^2 + 5*y^2 = n.

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%F G.f.: Product_{k>=1} (1 + x^(4*k-2))^2*(1 - x^(4*k))*(1 + x^(10*k-5))^2*(1 - x^(10*k)).

%e G.f. = 1 + 2*q^2 + 2*q^5 + 4*q^7 + 2*q^8 + 4*q^13 + 2*q^18 + 2*q^20 + 4*q^22 + ...

%t nmax = 98; CoefficientList[Series[EllipticTheta[3, 0, q^2] EllipticTheta[3, 0, q^5], {q, 0, nmax}], q]

%t nmax = 98; CoefficientList[Series[QPochhammer[-q^2, -q^2] QPochhammer[-q^5, -q^5]/(QPochhammer[q^2, -q^2] QPochhammer[q^5, -q^5]), {q, 0, nmax}], q]

%Y Cf. A000286, A020674, A033718, A106889, A108563, A192323.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 02 2018