help with a new sequence

[Emeric Deutsch]

Seqfans,

I intend to submit to OEIS the following new sequence (triangle):

1,1,1,1,3,1,1,4,2,6,1,1,5,5,10,10,10,1,

Triangle read by rows: T(n,k) is the number of Dyck paths of
semilength n whose ascent lengths form the k-th partition of
the integer n; the partitions of n are ordered in the way
exemplified by [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1],
[1,1,1,1,1] (the "Mathematica" ordering).

Equivalently, T(n,k) is the number of ordered trees with n
edges whose node degrees form the k-th partition of the
integer n.

Example: T(5,3)=5 because the 3rd partition of 5 is [3,2] and
we have (UU)DD(UUU)DDD, (UUU)DDD(UU)DD, (UU)D(UUU)DDDD,
(UUU)D(UU)DDDD, and (UUU)DD(UU)DDD; here U=(1,1), D=(1,-1) and
the ascents are shown between parentheses.

Triangle starts:
1;
1,1;
1,3,1;
1,4,2,6,1;
1,5,5,10,10,10,1;

Row n has A000041(n) terms (=number of partitions of n).

------------

So far this is all I have; terms have been found by
straightforward counting. I am sure you can find more
terms, more facts, etc. I'd appreciate any collaboration.

Thanks, Emeric