Delannoy Number
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), andnortheast (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 valuesof 
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 CatalanNumbers do to the Binomial Coefficients.
See also Binomial Coefficient, Catalan Number, Motzkin Number, Schröder Number
References
Moser, L. ``King Paths on a Chessboard.'' Math. Gaz. 39, 54, 1955. Sloane, N. J. A. SequenceA001850/M2942in ``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.
- Pine Cone Number
- Piriform
- Pisot-Vijayaraghavan Constants
- Pistol
- Pitchfork Bifurcation
- Pivoting
- Pivot Theorem
- Pizza Theorem
- Place (Digit)
- Place (Field)
- Place (Game)
- Planar Bubble Problem
- Planar Distance
- Planar Graph
- Planar Point
- Planar Space
- Planar Straight Line Graph
- Plancherel's Theorem
- Planck's Radiation Function
- Plane
- Plane Curve
- Plane Cutting
- Plane Division
- Plane-Filling Curve
- Plane-Filling Function