For $d=1$, let $M_1 = 1$. For $d>1$, define $M_d$ recursively by $M_d = d(d!) - \sum_{i=1}^{d-1} (d-i)! M_i.$
Is $M_d$ bounded by a polynomial (of some high degree) in $d$?
Note that $M_d$ is the number of subgroups of index $d$ in the free group of rank $2$.