Counting Lattice Paths With Four Types of Steps

The main purpose of this paper is to derive generating functions for the numbers of lattice paths running from (0, 0) to any (n, k) in Z×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z} \times \mathbb{N}}$$\end{document} consisting of four types of steps: horizontal H = (1, 0), vertical V = (0, 1), diagonal D = (1, 1), and sloping L = (–1, 1). These paths generalize the well-known Delannoy paths which consist of steps H, V, and D. Several restrictions are considered. However, we mainly treat with those which will be needed to get the generating function for the numbers R(n, k) of these lattice paths whose points lie in the integer rectangle {(x,y)∈N2:0≤x≤n,0≤y≤k}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{(x, y) \in \mathbb{N}^2 : 0 \leq x \leq n, 0 \leq y \leq k\}}$$\end{document}. Recurrence relation, generating functions and explicit formulas are given. We show that most of considered numbers define Riordan arrays.