[seqfan] Observation on A080936

[David Wilson]

For A080936, T(n,k) also counts the number of balanced stings of n pairs 
parentheses having nesting depth k.

For example, for 3 pairs of parentheses, there are 5 balanced strings

()()(), ()(()), (()()), (())(), ((()))

with nesting depths 1, 2, 2, 2, 3 respectively, so that

T(3,1) = 1
T(3,2) = 3
T(3,3) = 1

I think that there is probably a simple bijection to Dyck paths that 
makes this clear.

The current table is

1;
1,1;
1,3,1;
1,7,5,1;
1,15,18,7,1;
1,31,57,33,9,1;
1,63,169,132,52,11,1;

I think a zero row and column should be added to give

1;
0,1;
0,1,1;
0,1,3,1;

&c.