Parentheses Inventor

You invent a new language that has $k \geq 1$ forms of parentheses, say $(_1 )_1$ to $(_k )_k$. A proper $(k,n)-$parenthesization is a string of length $2n$ such that for all $1 \leq i \leq k$, the number of $)_i$ never exceeds the number of $(_i$ anywhere in the string. Find the number of proper $(k,n)-$parenthesizations when $k = 4$ and $n = 5$.