Closed form for the recursion $\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$

Question

I was completing a computer science problem when the following recursion popped up:

$u_0=1$

$\displaystyle u_n=\sum_{k=0}^{n-1} u_ku_{n-1-k}$

Is there a closed form for this recursion ?

I can't find anything on my own.

Answer 1

Yes, there is a closed form for this famous sequence:

$$u_n = \frac{1}{n+1}\binom{2n}{n}.$$