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Number of paths of length 2n+2

Source: Czech and Slovak Olympiad 2015, National Round, Problem 2

April 1, 2015
combinatoricslattice paths

Problem Statement

Let A=[0,0]A=[0,0] and B=[n,n]B=[n,n]. In how many ways can we go from AA to BB, if we always want to go from lattice point to its neighbour (i.e. point with one coordinate the same and one smaller or bigger by one), we never want to visit the same point twice and we want our path to have length 2n+22n+2? (For example, path [0,0],[0,1],[−1,1],[−1,2],[0,2],[1,2],[2,2],[2,3],[3,3][0,0],[0,1],[-1,1],[-1,2],[0,2],[1,2],[2,2],[2,3],[3,3] is one of the paths for n=3n=3)
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