The Catalan numbers $C_n$ count the number of staircase walks from $(0,0)$ to $(n,n)$ that lie below the diagonal $y=x$. Let $C(n,k)$ denote the number of staircase walks from $(k,0)$ to $(n,n)$ that lie below the diagonal $y=x$. E.g. $C(n,0)=C_n,\ C(n,n)=1,\ C(n,n-1)=n-1$.
I would like to know if there is a nice formula, e.g. $C_n=\frac{1}{n+1}{2n\choose n}$ for the usual Catalan numbers, that applies to these generalized Catalan numbers.