Seqfans,
      Row n of a related triangle could count the number of terms in row n 
of Emerics triangle that sum to form the respective terms of the Narayana triangle. 
 
This triangle P would begin:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 2, 2, 1, 1; ...

where row sums equal the partition numbers.
 
For example, row 3 of P is [1, 2, 1, 1]
since Narayana row 3 is [1, 4+2, 6, 1] 
formed from row 3 of Emerics triangle: [1, 4, 2, 6, 1].
 
Is the resulting partition-related triangle P already in the OEIS?

It might have an interesting matrix inverse. 
    Paul
 
-- Emeric Deutsch <deutsch@duke.poly.edu> wrote:
...
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).
...
Thanks, Emeric

