  We consider random normal matrix and planar symplectic ensembles, which can
be interpreted as two-dimensional Coulomb gases having determinantal and
Pfaffian structures, respectively. For general radially symmetric potentials,
we derive the asymptotic expansions of the log-partition functions up to and
including the $O(1)$-terms as the number $N$ of particles increases. Notably,
our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in
these expansions depend on the connectivity of the droplet. For random normal
matrix ensembles, our formulas agree with the predictions proposed by Zabrodin
and Wiegmann up to a universal additive constant. For planar symplectic
ensembles, the expansions contain a new kind of ingredient in the $O(N)$-terms,
the logarithmic potential evaluated at the origin in addition to the entropy of
the ensembles.
