In mathematics, a Delannoy number ${\displaystyle D}$ counts the paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.

The Delannoy number ${\displaystyle D(m,n)}$ also counts the global alignments of two sequences of lengths ${\displaystyle m}$ and ${\displaystyle n}$, the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the cells in an m-dimensional von Neumann neighborhood of radius n.

The Delannoy number D(3, 3) equals 63. The following figure illustrates the 63 Delannoy paths from (0, 0) to (3, 3):

The subset of paths that do not rise above the SW–NE diagonal are counted by a related family of numbers, the Schröder numbers.

The Delannoy array is an infinite matrix of the Delannoy numbers:

In this array, the numbers in the first row are all one, the numbers in the second row are the odd numbers, the numbers in the third row are the centered square numbers, and the numbers in the fourth row are the centered octahedral numbers. Alternatively, the same numbers can be arranged in a triangular array resembling Pascal's triangle, also called the tribonacci triangle, in which each number is the sum of the three numbers above it:

The row sums of the triangle give successive Pell numbers 1, 2, 5, 12, 29, 70, 169,... the sums of the rising diagonals give the tribonacci numbers 1, 1, 2, 4, 7, 13, 24,... 

The central Delannoy numbers D(n) = D(n, n) are the numbers for a square n × n grid. The first few central Delannoy numbers (starting with n = 0) are: