The $n$-th Catalan number $c_n$ has the closed form $\frac1{n+1}\binom{2n}{n}$ and follows the recursion $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$
I am interested in the quantity $e_n$ which follows the recursion $e_n= (n-1) \sum\limits_{i = 1}^{n-1}{e_i e_{n-i}}$ for $n > 1$, with $e_1 = 1$.
I am wondering if it is possible to approximate $e_n$ using $c_n$?