I am studying the recurrence relation
$x(n) = \sum_{i=1}^{n-1} x(i) x(n-i), \hspace{20px} x(1) = 1$
and have found without proof the explicit solution
$x(n) = \frac{(2n-2)!}{n!(n-1)!}, \hspace{20px} n \in \mathbb{N}$
Do you have a clever way to verify that this is indeed a solution? I can not find a way to erase the sum when inserting the expression in the relation.