Catalan numbers in a series expansion of the directed percolation probability on a square lattice

We regard the bond directed percolation on a square lattice as a discrete-time Markov process of a one-dimensional interacting particle system. The coefficients in series expansion of the probability P-n,P-m of having m particles at time n - 1 are studied. We derive the difference equations for the first and the second series of coefficients and prove that these coefficients are expressed using the ballot numbers, whose special cases are known as the Catalan numbers. As a corollary of our results, we prove a part of the conjecture by Baxter and Guttmann that the correction terms are expressed as rational functions of the Catalan numbers. We also give approximations for the percolation probability using the present formula.