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A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

%I A302860 #10 Feb 16 2025 08:33:53
%S A302860 1,3,9,27,89,333,1341,5449,21697,84663,327829,1275739,5020457,
%T A302860 19964623,79883141,320317827,1284656385,5152761033,20686311261,
%U A302860 83182322509,335110196569,1352277390001,5463873556381,22097867887045,89441286136465,362277846495883,1468465431530457
%N A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.
%C A302860 a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).
%H A302860 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A302860 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A302860 a(n) = A122510(n,n).
%F A302860 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 A302860 Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
%t A302860 Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]
%Y A302860 Main diagonal of A122510.
%Y A302860 Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.
%K A302860 nonn
%O A302860 0,2
%A A302860 _Ilya Gutkovskiy_, Apr 14 2018