Prove this recurrence relation? (catalan numbers)

Question

$$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots + C_kC_{n−k} + \cdots + C_nC_0\text{ ?}$$

Where $C_n$ denotes the number of ways of writing a valid list of open and closed parentheses of length $2n$?

Answer 1

$$(1+x)^m(x+1)^m=(1+x)^{2m}$$

The coefficients of $x^m$ $$\sum_{r=0}\binom mr\binom m{m-r}=\binom{2m}m$$