For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
Show by a combinatorial argument that, for every integer $k \geqslant 1$,
$$C_{k} = \sum_{i=0}^{k-1} C_{i} C_{k-i-1}$$
We can note that a well-parenthesized word is necessarily of the form $(m)m^{\prime}$ with $m$ and $m^{\prime}$ two well-parenthesized words, possibly empty.