  We obtain an explicit formula for a certain weighted enumeration of lozenge
tilings of a hexagon with an arbitrary triangular hole. The complexity of our
expression depends on the distance from the hole to the center of the hexagon.
This proves and generalizes conjectures of Ciucu et al., who considered the
case of plain enumeration when the triangle is located at or very near the
center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete
and continuous Selberg-type integrals.
