The Delannoy number D(n, m) is the number of paths from (0, 0) to (n, m) in the grid, where each step is either right or up. This is a classic problem in combinatorics and has many applications in counting problems.

The recurrence relation for the Delannoy numbers is D(n, m) = D(n - 1, m) + D(n, m - 1). This means that the Delannoy number D(n, m) can be computed by adding the Delannoy numbers D(n - 1, m) and D(n, m - 1).

The generating function for the Delannoy numbers is F(x, y) = (1 - x - y)^-1. This means that the Delannoy number D(n, m) can be computed by taking the coefficient of x^n y^m in the expansion of F(x, y).

The asymptotic formula for the Delannoy numbers is D(n, m) ~ (2 / sqrt(pi)) * n^(3/2) * m^(3/2). This means that for large values of n and m, the Delannoy number D(n, m) is approximately equal to (2 / sqrt(pi)) * n^(3/2) * m^(3/2).

The Delannoy numbers have a wide variety of applications in combinatorics, including counting lattice paths, counting tilings, counting permutations, and counting graphs.