  We begin by considering a sequence of polynomials in three variables whose
coefficients count restricted binary overpartitions with certain properties. We
then concentrate on two specific subsequences that are closely related to the
Chebyshev polynomials of both kinds, deriving combinatorial and algebraic
properties of some special cases. We show that the zeros of these polynomial
sequences lie on certain algebraic curves, some of which we study in greater
detail. Finally, we extend part of this work to restricted $b$-ary
overpartitions for arbitrary integers $b\geq 2$.
