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Expansion of (1 + theta_3(q))^2*(1 + theta_3(q^2))^2/16, where theta_3() is the Jacobi theta function.
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%I #5 Feb 16 2025 08:33:56

%S 1,2,3,4,5,4,5,4,5,8,8,8,11,8,6,8,5,10,14,12,16,12,11,8,11,14,14,20,

%T 18,12,14,12,5,20,19,20,30,16,17,16,16,18,24,20,25,28,14,16,11,22,25,

%U 28,34,20,30,24,18,28,26,28,42,24,20,32,5,28,36,28,41,32,32,20,30,30,28,44

%N Expansion of (1 + theta_3(q))^2*(1 + theta_3(q^2))^2/16, where theta_3() is the Jacobi theta function.

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

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

%e G.f. = 1 + 2*q + 3*q^2 + 4*q^3 + 5*q^4 + 4*q^5 + 5*q^6 + 4*q^7 + 5*q^8 + ...

%t nmax = 75; CoefficientList[Series[(1 + EllipticTheta[3, 0, q])^2 (1 + EllipticTheta[3, 0, q^2])^2/16, {q, 0, nmax}], q]

%t nmax = 75; CoefficientList[Series[(1 + QPochhammer[-q, -q]/QPochhammer[q, -q])^2 (1 + QPochhammer[-q^2, -q^2]/QPochhammer[q^2, -q^2])^2/16, {q, 0, nmax}], q]

%Y Cf. A014110, A097057, A317645.

%K nonn,changed

%O 0,2

%A _Ilya Gutkovskiy_, Aug 02 2018