Moments of generalized Motzkin paths

Consider lattice paths in the plane allowing the steps (1,1), (1,-1), and (w,0), for some nonnegative integer w. For n > 1, let E(n,0) denote the set of paths from (0,0) to (n,0) running strictly above the x-axis except initially and finally. Generating functions are given for sums of moments of the ordinates of the lattice points on the paths in E(n,0). In particular, recurrencess are derived for the cardinality, the sum of the first moments (essentially the area), and the sum of the second moments for paths in E(n,0). These recurrences unify known results for w= 0, 1, 2, i.e. those for the Dyck (or Catalan), Motzkin, and Schröder paths, respectively. The sum of the second moments is seen to equal the number of unrestricted paths running from (0,0) to (0,n-2).