Underdiagonal lattice paths with unrestricted steps

We use some combinatorial methods to study underdiagonal paths (on the Z(2) lattice) made up of unrestricted steps, i.e., ordered pairs of non-negative integers. We introduce an algorithm which automatically produces some counting generating functions for a large class of these paths. We also give an example of how we use these functions to obtain some specific information on the number d(n,k) of paths from the origin to a generic point (n, n - k).