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A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

%I A302997 #14 Feb 16 2025 08:33:53
%S A302997 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,33,29,9,1,1,11,89,123,49,11,1,1,
%T A302997 13,221,425,257,81,13,1,1,15,485,1343,1281,515,113,15,1,1,17,953,4197,
%U A302997 5913,3121,925,149,17,1,1,19,1713,12435,23793,16875,6577,1419,197,19,1,1,21,2869,33809,88273,84769,42205,11833,2109,253,21,1
%N A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.
%C A302997 A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.
%H A302997 Andrew Howroyd, <a href="/A302997/b302997.txt">Table of n, a(n) for n = 0..1274</a>
%H A302997 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A302997 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A302997 A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j=-infinity..infinity} x^(j^2))^k.
%e A302997 Square array begins:
%e A302997   1,   1,   1,    1,     1,      1,  ...
%e A302997   1,   3,   5,    7,     9,     11,  ...
%e A302997   1,   5,  13,   33,    89,    221,  ...
%e A302997   1,   7,  29,  123,   425,   1343,  ...
%e A302997   1,   9,  49,  257,  1281,   5913,  ...
%e A302997   1,  11,  81,  515,  3121,  16875,  ...
%t A302997 Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k/(1 - x), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t A302997 Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A302997 (PARI) T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ _Andrew Howroyd_, Sep 14 2019
%Y A302997 Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
%Y A302997 Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
%Y A302997 Main diagonal gives A302861.
%Y A302997 Cf. A000122, A122510, A302996, A302998.
%K A302997 nonn,tabl
%O A302997 0,5
%A A302997 _Ilya Gutkovskiy_, Apr 17 2018