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%I #12 Mar 27 2023 17:48:36
%S 1,1,0,1,2,0,1,4,2,0,1,6,4,2,0,1,8,6,4,2,0,1,10,24,30,4,2,0,1,12,90,
%T 104,6,12,2,0,1,14,252,250,24,30,4,2,0,1,16,574,876,730,248,30,4,2,0,
%U 1,18,1136,3542,4092,1210,312,54,4,2,0,1,20,2034,12112,18494,7812,2250,456,6,4,2,0
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k, where theta_3() is the Jacobi theta function.
%C A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.
%H Alois P. Heinz, <a href="/A302996/b302996.txt">Antidiagonals n = 0..200, flattened</a>
%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,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, 10, ...
%e 0, 2, 4, 6, 24, 90, ...
%e 0, 2, 4, 30, 104, 250, ...
%e 0, 2, 4, 6, 24, 730, ...
%e 0, 2, 12, 30, 248, 1210, ...
%p b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
%p b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
%p end:
%p A:= (n, k)-> b(n^2, k):
%p seq(seq(A(n,d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 10 2023
%t Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t Table[Function[k, SeriesCoefficient[Sum[x^i^2, {i, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%Y Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
%Y Main diagonal gives A232173.
%Y Cf. A000122, A122141, A255212, A286815.
%K nonn,tabl
%O 0,5
%A _Ilya Gutkovskiy_, Apr 17 2018
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