Let's start the question with a prototype recursion of the numbers $a_g(d)$ we are interested in. These numbers satisfy the following recursion where $g\geq 0, d\geq 1$ all the numbers are otherwise zero.
$$ d a_g (d)-2 (2d-3)a_g (d-1)-d(d-1)^2 a_{(g-1)}(d)=0 .$$
Can we put some upper bound for the $a_g (d)$ probably of the from $ c^g s^n (2g-kd+r)!$ where $c,ks,r$ are constant? How does one come up with the bound steps?
In general, if we have a recursion of the from
$$p_r (d) a_{d+r} + p_{r-1} a_{d+r-1}+\cdots +p_{0}a_{0}=0 $$
say we now we fix the order of the above recursion that is given by $\alpha$ and degree of the above recursion that is given by $\beta$ then can we come up with a similar bound with some constant exponential and by $(2g-kd+r)!$?
Any literature regarding the above is very welcomed.