%I #9 Apr 15 2018 02:55:55

%S 1,3,9,27,89,333,1341,5449,21697,84663,327829,1275739,5020457,

%T 19964623,79883141,320317827,1284656385,5152761033,20686311261,

%U 83182322509,335110196569,1352277390001,5463873556381,22097867887045,89441286136465,362277846495883,1468465431530457

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

%C a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(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>

%F a(n) = A122510(n,n).

%F a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - _Vaclav Kotesovec_, Apr 14 2018

%t Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]

%t Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

%Y Main diagonal of A122510.

%Y Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 14 2018