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%I #60 Jul 17 2020 12:17:44
%S 1,1,0,1,2,0,1,4,0,0,1,6,4,0,0,1,8,12,0,2,0,1,10,24,8,4,0,0,1,12,40,
%T 32,6,8,0,0,1,14,60,80,24,24,0,0,0,1,16,84,160,90,48,24,0,0,0,1,18,
%U 112,280,252,112,96,0,4,2,0,1,20,144,448,574,312,240,64,12
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.
%C A(n,k) is the number of ways of writing n as a sum of k squares.
%C This is the transpose of the array in A122141.
%H Seiichi Manyama, <a href="/A286815/b286815.txt">Antidiagonals n = 0..139, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sum_of_squares_function">Sum of squares function</a>
%F G.f. of column k: (Product_{j>=1} (1 - x^(2*j))^5/((1 - x^j)*(1 - x^(4*j)))^2)^k.
%e Square array begins:
%e 1, 1, 1, 1, 1, ...
%e 0, 2, 4, 6, 8, ...
%e 0, 0, 4, 12, 24, ...
%e 0, 0, 0, 8, 32, ...
%e 0, 2, 4, 6, 24, ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
%p A(n, k-1) +2*add(A(n-j^2, k-1), j=1..isqrt(n))))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 27 2017
%t A[n_, k_] := A[n, k] = If[n == 0, 1, If[n < 0 || k < 1, 0, A[n, k-1] + 2*Sum[A[n-j^2, k-1], {j, 1, Sqrt[n]}]]];
%t Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 28 2018, after _Alois P. Heinz_ *)
%Y Columns k=0-16 give: A000007, A000122, A004018, A005875, A000118, A000132, A000141, A008451, A000143, A008452, A000144, A008453, A000145, A276285, A276286, A276287, A000152.
%Y Diagonal gives A066535.
%Y Cf. A122141.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, May 27 2017
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