Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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