I was playing around with an area of graph theory and was able to recursively define the number of graphs $N$ that fit a specific condition dependent on two variables, $d,v\in\mathbb{N_0}$, where
$$N(0,v)=v^2,$$ $$N(d,v)=N(d-1,v+1)+N(d-1,v)\big(N(d-1,v)+1\big).$$
I would like to deduce the asymptotic nature of $N(d,0)$, or at the very least, some upper and lower bounds.
Just based off computational estimates for $0\leq d\leq 25$, it seems quite likely that
$$\log_2\Big(1+\log_2\big(1+N(d,0)\big)\Big)\sim d.$$
Which, due to $N$ being defined recursively, unsurprisingly indicates that $N(d,0)$ grows hyper-exponentially. However, this is the first time I've come across naturally, let alone dealt with, a recursive function of two-variables. So, are there any theorems or tools I can use to deal with getting information about the growth rate of $N$?