Counting Proper Parenthesizations (Catalan Numbers)
A *proper parenthesization* of length
n$ is a string of $n$ left parentheses and $n$ right parentheses such that at every prefix, the number of right parentheses never exceeds the number of left parentheses.
For example, with $n = 3$: the strings $()()()$, $((()))$, and $()(())$ are valid, while $())(()$ and $(((())$ are not.
How many proper parenthesizations of length