Area of Catalan paths

A Catalan path of length n, is a path from (0,0) going N=(0,1) or E=(0,1) in each step, such that it stays under (or on) the line x=y and ending at (n,n) after 2n steps. Here we calculate the total area between the line y=x and all Catalan paths. The total area an=12∑k=1nck4n−k, where ck is the number of Catalan paths of length k.