The Delannoy numbers are defined by

where . They are the number of lattice paths from to in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e, , , and ).

For , the Delannoy numbers are the number of ``king walks''

where is a Legendre Polynomial (Moser 1955, Vardi 1991). Another expression is

where is a Binomial Coefficient and is a Hypergeometric Function. The values of for , 2, ... are 3, 13, 63, 321, 1683, 8989, 48639, ... (Sloane's A001850).

The Schröder Numbers bear the same relation to the Delannoy numbers as the Catalan Numbers do to the Binomial Coefficients.

**References**

Moser, L. ``King Paths on a Chessboard.'' *Math. Gaz.* **39**, 54, 1955.

Sloane, N. J. A. Sequence
A001850/M2942
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Vardi, I. *Computational Recreations in Mathematica.* Reading, MA: Addison-Wesley, 1991.

© 1996-9

1999-05-24