Question 133
Let $\mathbb{N} = \{0,1,2,\dots\}$ be the set of all non-negative integers. For each $n \in \mathbb{N}$, define the Catalan number
$$
C_n = \frac{1}{n+1} \binom{2n}{n} = \frac{(2n)!}{n!(n+1)!}.
$$
Prove that for any positive integer $m$, we have
$$
\sum_{\substack{i,j,k \in \mathbb{N} \\ i+j+k=m}} C_{i+j}C_{i+k}C_{j+k} = \frac{3}{2m+3}C_{2m+1}.
$$
$$
C_n = \frac{1}{n+1} \binom{2n}{n} = \frac{(2n)!}{n!(n+1)!}.
$$
Prove that for any positive integer $m$, we have
$$
\sum_{\substack{i,j,k \in \mathbb{N} \\ i+j+k=m}} C_{i+j}C_{i+k}C_{j+k} = \frac{3}{2m+3}C_{2m+1}.
$$