Delannoy number

In mathematics, a Delannoy number ${\displaystyle D}$ describes the number of 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.

Delannoy number

Named after	Henri–Auguste Delannoy
No. of known terms	infinity
Formula	${\displaystyle D(m,n)=\sum _{k=0}^{\min(m,n)}{\binom {m}{k}}{\binom {n}{k}}2^{k}}$
OEIS index	A008288 Square array of Delannoy

The Delannoy number ${\displaystyle D(m,n)}$ also counts the number of global alignments of two sequences of lengths ${\displaystyle m}$ and ${\displaystyle n}$, the number of points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the number of cells in an m-dimensional von Neumann neighborhood of radius n while the number of cells on a surface of an m-dimensional von Neumann neighborhood of radius n is given with (sequence A266213 in the OEIS).