%I #9 Mar 26 2015 14:45:06
%S 1,1,1,1,3,3,1,7,15,15,1,15,57,105,105,1,31,195,561,945,945,1,63,633,
%T 2685,6555,10395,10395,1,127,1995,12105,40725,89055,135135,135135,1,
%U 255,6177,52605,237555,684495,1381905,2027025,2027025,1,511,18915,223161
%N Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.
%H J. L. Martin, <a href="http://www.math.umn.edu/math/slopes.pdf">The slopes determined by n points in the plane</a>.
%H Martin, Jeremy L. <a href="http://arxiv.org/abs/math.AG/0302106">The slopes determined by n points in the plane</a>. Duke Math. J. 131 (2006), no. 1, 119-165 (also arXiv math.AG/0302106, but beware errors).
%F Recurrence: d(n, k) = 1 for n=0 or k=1, d(n, k) = 0 for n>0 and either k<0 or k>n, d(n, k) = d(n-1, k) + d(n, k-1) + sum( binomial(k-1, w)*binomial(n-k, x)*d(n-k+w-x, w+1)*d(k-w+x, k-w-1), w=0..k-2, x=0..n-k-1)
%e Triangle begins
%e 1,
%e 1, 1,
%e 1, 3, 3,
%e 1, 7, 15, 15,
%e 1, 15, 57, 105, 105,
%e 1, 31, 195, 561, 945, 945,
%e 1, 63, 633, 2685, 6555, 10395, 10395,
%e ...
%Y First three columns are A000012, A000225, A210448, rightmost two diagonals are both A001147, difference of second- and third-rightmost diagonals is A000165.
%K easy,nonn,tabl
%O 1,5
%A Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003