Average number of Dyck words in a Dyck word Given a integer $n$, how many Dyck words are a substring of a Dyck word of size $n$, on average?
For example, if $n=2$, then Dyck words of size $2$ are : 


*

*[ ] [ ]  

*[ [ ] ]


(1) contains two strict "sub-Dyck words" : [ ] (with the first two parentheses) and [ ] (with the last two parentheses). And the original [ ] [ ]. The total is 3
(2) contains only one strict "sub-Dyck word": [ ]. And the original [ [ ] ]. The total is 2
So for $n=2$ the answer is 2.5, of course it is harder to compute when $n$ gets bigger.
Has anyone an idea on how to find a general formula for this problem?

So, I've coded a little Python program that computes for each $n$, the total number of "sub-Dyck words" in all Dyck words of semi-length $n$
Here is the output for $n$ ranging from 1 to 13 : 1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860.
Which is know as A002054 in oeis. And that's even comment number 7. Bingo!
So the number I was looking for is $\frac{(2n+1)\times n}{n+2}$
Yet I don't consider the question solved, as I haven't found the proof for it.
