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%I #37 Jul 31 2017 21:03:26
%S 1,1,0,1,1,0,1,2,0,0,1,3,1,1,0,1,4,3,2,0,0,1,5,6,4,2,0,0,1,6,10,8,6,0,
%T 1,0,1,7,15,15,13,3,3,0,0,1,8,21,26,25,12,6,2,0,0,1,9,28,42,45,31,14,
%U 9,0,0,0,1,10,36,64,77,66,35,24,3,2,1,0,1,11,45
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.
%C A(n, k) is the number of ways of writing n as the sum of k triangular numbers.
%H Seiichi Manyama, <a href="/A286180/b286180.txt">Antidiagonals n = 0..139, flattened</a>
%F G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 0, 1, 3, 6, 10, ...
%e 0, 1, 2, 4, 8, 15, ...
%e 0, 0, 2, 6, 13, 25, ...
%t Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* _Michael De Vlieger_, May 07 2017 *)
%Y Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.
%Y Main diagonal gives A106337.
%K nonn,tabl
%O 0,8
%A _Seiichi Manyama_, May 07 2017
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