Abstract: We give the generating function, by perimeter, of two previously
unsolved combinatorial models. They are "3-choice polygons" and staircase (or
parallelogram) polygons with a staircase hole. The solutions are expressed as
high order holonomic or differentiably finite functions, given as the
solution of high order ODEs with polynomial coefficients.

These ODEs are found to be Fuchsian, but analysis of the nature of the
singularities from the indicial equation poses some challenges. We have
developed an algorithm for singularity analysis that greatly simplifies the
analysis.

The talk will hopefully be of interest to two groups -- those interested in
combinatorial models will be interested in the first part, while those
interested in scientific computing will be interested in the second part.
Hopefully, the first part will provide some motivation for the second.