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A302996 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.

%I A302996 #13 Feb 16 2025 08:33:53
%S A302996 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 A302996 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 A302996 1,18,1136,3542,4092,1210,312,54,4,2,0,1,20,2034,12112,18494,7812,2250,456,6,4,2,0
%N A302996 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 A302996 A(n,k) is the number of ordered ways of writing n^2 as a sum of k squares.
%H A302996 Alois P. Heinz, <a href="/A302996/b302996.txt">Antidiagonals n = 0..200, flattened</a>
%H A302996 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A302996 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A302996 A(n,k) = [x^(n^2)] (Sum_{j=-infinity..infinity} x^(j^2))^k.
%e A302996 Square array begins:
%e A302996   1,  1,   1,   1,    1,     1,  ...
%e A302996   0,  2,   4,   6,    8,    10,  ...
%e A302996   0,  2,   4,   6,   24,    90,  ...
%e A302996   0,  2,   4,  30,  104,   250,  ...
%e A302996   0,  2,   4,   6,   24,   730,  ...
%e A302996   0,  2,  12,  30,  248,  1210,  ...
%p A302996 b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
%p A302996       b(n, t-1)+2*add(b(n-j^2, t-1), j=1..isqrt(n))))
%p A302996     end:
%p A302996 A:= (n, k)-> b(n^2, k):
%p A302996 seq(seq(A(n,d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Mar 10 2023
%t A302996 Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t A302996 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 A302996 Columns k=0..4,7 give A000007, A040000, A046109, A016725, A267326, A361695.
%Y A302996 Main diagonal gives A232173.
%Y A302996 Cf. A000122, A122141, A255212, A286815.
%K A302996 nonn,tabl
%O A302996 0,5
%A A302996 _Ilya Gutkovskiy_, Apr 17 2018