I have the following recurrence relation for some coefficients
$$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$
with $b_1$ to $b_3$ and $P_0$ being the initial conditions of the series.
The problem is that I have to calculate the series to a high value of $n$ (about 10000, or sometimes even more). And since the $n$th term depends on a sum that involves all the other terms before it, the calculation is really slow. So I'm trying to simplify it in some way to make it faster to program. (Getting rid of the sum would be already a good start for example.)
The best cases scenario would be to have $b_n = F(n)$, which of course would be simpler to read and way faster to program, but I'm not sure this is possible. Is there a way to re-write the relation in a way that's faster to calculate? (Or at the very least in a simpler way?)
If you prove that it isn't possible to achieve what I want I'd be also extremely grateful. That way I can stop worrying about it.
Thank you.