An interesting recursive relation has originated in the solution to an Oxford MAT paper (2018 paper, Q5 (iv)).
The solution gives the correct recursive relation as $$t_n = \sum_{k=1}^{n-1}t_k t_{n-k},\qquad t_1 = 1.$$
I understand the intuition of how this is constructed, however, what I would like to know is whether there is a suitable method for constructing an overall expression for $t_n$ in terms of $t_1$ (or even $n$) only. If you start from $t_n$ and work your way backwards while using the recursive relation, you'll end up with a large number of repeated summations, which becomes very hard to keep track of (for me at least).
Is there some sort of neat way to obtain the appropriate expression for $t_n$ in the way I have described? It is very straightforward to start with the base case and show that $t_n = A_n t_1 ^ n$, where $A_n$ are numbers from the sequence $1,1,2,5,14,42,\ldots$, which, according to Wolfram Alpha, are given by
$$ A_n = \dfrac{4^{n-1}\Gamma(n-\tfrac{1}{2})}{\sqrt{\pi}\Gamma(n+1)},$$
where $\Gamma(n)$ is the Gamma function.