If $C\_n$ is the $n$th Catalan number, then show that they satisfy the following recurrence:

$$C\_0 = 1,\quad C\_{n+1} = C\_0C\_n + C\_1C\_{n−1}+ \cdots + C\_kC\_{n−k} + \cdots + C\_nC\_0\text{ ?}$$

I tried using a bijective proof with $n+1$ appearing in the denominator of the formula $C\_n$.

Also, I need to show that $C\_n$ determines the number of paths in the plane from
$(0, 0)$ to $(n, n) \in\mathbb N\times\mathbb N$, that don’t cross the main diagonal ($y = x$) if each step in the path is of the form $(1, 0)$, or $(0, 1)$. 

Any suggestions would help!

Edit: $C\_n$ denotes the valid list of open and closed parentheses of length $2n$
(valid meaning the number of open parentheses is greater than or equal to closed parentheses)