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a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.
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%I #6 Apr 15 2018 13:28:32

%S 1,2,4,8,20,57,160,422,1076,2780,7449,20462,56348,153909,418268,

%T 1139703,3126068,8618611,23801146,65708424,181391905,501296216,

%U 1387834518,3848187985,10680579812,29660831057,82415406493,229156296047,637659848888,1775648562970,4947475298595

%N a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

%C a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n.

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

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%t Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n}], {n, 0, 30}]

%t Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n}], {n, 0, 30}]

%Y Cf. A000606, A003059, A010052, A224212, A224213, A287617, A302860, A302863.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 14 2018