  We derive a generating series for the number of free subgroups of finite
index in $\Delta^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between
free subgroups of $\Delta^+$ and certain hypermaps (also known as ribbon graphs
or "fat" graphs), and show that this generating series is transcendental. We
provide non-linear recurrence relations for the above numbers based on
differential equations that are part of the Riccati hierarchy. We also study
the generating series for conjugacy classes of free subgroups of finite index
in $\Delta^+$, which correspond to isomorphism classes of hypermaps. Asymptotic
formulas are provided for the numbers of free subgroups of given finite index,
conjugacy classes of such subgroups, or, equivalently, various types of
hypermaps and their isomorphism classes.
