I have come across this sequence
$$a_0 = -2, a_1 = 5, a_2 = -28, a_3 = 255$$
and, in general
$$a_n = -\frac{1}{2}\bigg(\sum_{i=1}^n \binom{2n+4}{2i}a_{n-i} + \binom{2n+4}{2n+1}\bigg)$$
I've tried finding a closed form for it and couldn't. I've also tried plugging it into the OEIS and it didn't return any results. I'm wondering if anyone has seen a sequence like this before or if anyone has any ideas for finding a closed form for it.