I have a quadratic recurrence equation $\forall n \ge 2$ of the form
$$ f(n)=\sum_{l=1}^{n-1} f(l)f(n-l), $$
with the initial condition $f(1) =1$. The first few terms of the series are given by
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, ...
I am interested in an upper bound for this equation. By means of trial and error (up to 10k terms) it turns out that the following upper bound works $$ f(n) < 2^{2n}. $$
How can one proof that this upper bound is indeed true for all $n$?