cp's OEIS Frontend

A079268 Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.

%I A079268 #11 Aug 05 2025 18:10:17
%S A079268 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 A079268 2685,6555,10395,10395,1,127,1995,12105,40725,89055,135135,135135,1,
%U A079268 255,6177,52605,237555,684495,1381905,2027025,2027025,1,511,18915,223161
%N A079268 Triangle read by rows: d(n,k) = number of decreasing labeled trees with n nodes and largest leaf <= k, for 1 <= k <= n.
%H A079268 J. L. Martin, <a href="http://www.math.umn.edu/math/slopes.pdf">The slopes determined by n points in the plane</a>.
%H A079268 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 A079268 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_{w=0..k-2, x=0..n-k-1} binomial(k-1, w) * binomial(n-k, x) * d(n-k+w-x, w+1) * d(k-w+x, k-w-1).
%e A079268 Triangle begins
%e A079268 1,
%e A079268 1, 1,
%e A079268 1, 3, 3,
%e A079268 1, 7, 15, 15,
%e A079268 1, 15, 57, 105, 105,
%e A079268 1, 31, 195, 561, 945, 945,
%e A079268 1, 63, 633, 2685, 6555, 10395, 10395,
%e A079268 ...
%Y A079268 First three columns are A000012, A000225, A210448, rightmost two diagonals are both A001147, difference of second- and third-rightmost diagonals is A000165.
%K A079268 easy,nonn,tabl
%O A079268 1,5
%A A079268 Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003