Delannoy number

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

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.