> 
> We define a sequence by $a\_0=0$ and $$a\_n=1+\sum\_{k=0}^n a\_k a\_{n-k}\quad n\ge1$$
>  Find a non-recursive formula for $a\_n$.
> 
> 
> 

Not homework, this is a question from my friend. I tried to use summation by parts
$$
\sum\_{k=m}^na\_kb\_k
=A\_nb\_n-A\_{m-1}b\_m-\sum\_{k=m}^{n-1}A\_k(b\_{k+1}-b\_k)
$$
But it doesn't make this problem easier. I am wondering if this problem is a special one ($a\_0=0$) and then it has a non-recursive formula. Any hints would be highly appreciated!