count all possible paths of length n in an undirected graph with dynamic programming

Given is an infinitely large grid graph. Use dynamic programming to calculate the number of possible paths of a given length n from a given start node, so that fjor every path applies: a) no vertex may be visited twice within it and b) never to go down an edge example of lattice graph for paths of length 2

I found out, that the number of possible paths corresponds to the row sums of the binomial coefficent, times 2 minus the center row, but if n is bigger than 3 other paths emerge which I struggle to include in my dynamic programming solution