Ballot number representation of the percolation probability series for the directed square lattice

Series expansion data are matters of increasing importance far studying the directed percolation problem and others which are not yet solved. In order to extrapolate series for the percolation probability on the directed square lattice, Baxter and Guttmann proposed a numerical method based on an assumption that the so-called correction terms are expressed as rational functions of the Catalan numbers. We give a theorem that the coefficients of the series are generally given as finite series of the ballot numbers, which proves the assumption by Baxter and Guttmann as a corollary. The proof of the theorem gives a method to calculate correction terms exactly, as demonstrated by calculating the first three correction terms explicitly. Although the present work provides a mathematical basis for the extrapolation procedure, there are still open problems concerning this procedure.