  We address the problem of enumerating paths in square lattices, where allowed
steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We
consider two such lattices differing in whether the (1,1) steps are allowed
along the diagonal itself. Our analysis leads to explicit generating functions
and an efficient way to compute terms of many sequences in the Online
Encyclopedia of Integer Sequences, proposed by Clark Kimberling almost two
decades ago.
