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%I #6 Sep 11 2012 07:21:30
%S 1,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,2,1,
%T 1,1,0,0,0,0,1,2,1,1,1,0,0,0,0,1,2,2,1,1,1,0,0,0,0,1,2,2,2,1,1,1,0,0,
%U 0,0,0,2,3,2,2,1,1,1,0,0,0,0,0,2,3,3,2,2,1,1,1,0,0,0,0,0,1,3,4,3,2,2,1,1,1
%N As an upper right triangle, number of distinct partitions of n where the highest part is k (0<=k<=n).
%F T(n, k) =sum_j[T(n-k, j)] for k>j with T(0, 0)=1
%e Rows are {1,0,0,0,...}, {1,0,0,0,...}, {1,1,0,0,...}, {1,1,1,1,...}, {1,1,1,2,...} etc. T(7,4)=2 since 7 can be written as 4+3 or 4+2+1. T(12,6)=3 since 12 can be written as 6+5+1 or 6+4+2 or 6+3+2+1.
%t t[n_?Positive, k_] := t[n, k] = Sum[t[n-k, j], {j, 0, k-1}]; t[0, 0] = 1; t[0, _] = 0; t[_?Negative, _] = 0; Table[ t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 11 2012 *)
%Y As upper right triangle, row sum is A011782, column sum is A000009, column maximum is A025591 (offset), row maximum is A026839 (offset). Cf. A026836 for this triangle starting at (1, 1) rather than (0, 0).
%K nonn,tabl
%O 0,33
%A _Henry Bottomley_, Jan 30 2001