Enumerating a class of lattice paths

Let script D sign₀(n) denote the set of lattice paths in the xy-plane that begin at (0,0), terminate at (n,n), never rise above the line y = x and have step set script S sign = {(k,0):k∈ℕ ⁺}∪{(0,k):k∈ℕ⁺}. Let ℰ₀(n) denote the set of lattice paths with step set script S sign that begin at (0,0) and terminate at (n,n). Using primarily the symbolic method (R. Sedgewick, P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996) and the Lagrange inversion formula we study some enumerative problems associated with script D sign₀(n) and ℰ₀(n).