I'm pretty sure this is A001263, a.k.a. the Narayana numbers, and that the formula is
T(n,k) = C(n-1,k-1) C(n,k-1)/k with 1<=k<=n
One interpretation of A001263 is that T(n,k) is the number of Dyck n-paths with exactly k peaks, and you can interpret each Dyck step as either (
or )
and each peak as ()
.