Euler Coefficients and Restricted Dyck Paths

We consider the problem of enumerating Dyck paths staying weakly above the x-axis with a limit to the number of consecutive up steps, or a limit to the number of consecutive down steps. We use Finite Operator Calculus to obtain formulas for the number of all such paths reaching a given point in the first quadrant. All our results are based on the Eulerian coefficients.