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.^[1]

The Delannoy number ${\displaystyle D(m,n)}$ also counts the global alignments of two sequences of lengths ${\displaystyle m}$ and ${\displaystyle n}$,^[2] the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin,^[3] and, in cellular automata, the cells in an m-dimensional von Neumann neighborhood of radius n.^[4]