The Narayan number 
for 
,
2, ... and 
,
..., 
gives a solution to several counting problems in combinatorics. For example, 
gives the number of expressions
with 
pairs of parentheses that are correctly matched and contain 
distinct nestings. It also gives the number Dyck
paths of length 
with exactly 
peaks.
A closed-form expression of 
is given by
 |
where 
is a binomial coefficient.
Summing over 
gives the Catalan number
 |
Enumerating 
as a number triangle is called the Narayana
triangle.
