Delannoy Number
The Delannoy numbers 
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 
). They are
given by the recurrence relation
 |
(1)
|
with 
. The are also given by the sums
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a hypergeometric
function.
A table for values for the Delannoy numbers is given by
 |
(5)
|
(OEIS A008288) for 
, 1, ... increasing
from left to right and 
, 1, ... increasing from top to bottom.
They have the generating function
 |
(6)
|
(Comtet 1974, p. 81).
Taking 
gives the central Delannoy numbers
, which are the number of "king walks"
from the 
corner of an 
square
to the upper right corner 
. These are
given by
 |
(7)
|
where 
is a Legendre
polynomial (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression
is
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
is a binomial
coefficient and 
is a hypergeometric
function. These numbers have a surprising connection with the Cantor
set (E. W. Weisstein, Apr. 9, 2006).
They also satisfy the recurrence equation
 |
(11)
|
They have generating function
 |  |  |
(12)
|
 |  |  |
(13)
|
The values of 
for 
, 2, ... are
3, 13, 63, 321, 1683, 8989, 48639, ... (OEIS A001850).
The numbers of decimal digits in 
for
, 1, ... are 1, 7, 76, 764, 7654, 76553, 765549,
7655510, ... (OEIS A114470), where the digits
approach those of 
(OEIS
A114491).
The first few prime Delannoy numbers are 3, 13, 265729, ... (OEIS A092830), corresponding to indices 1, 2, 8, ..., with no others for 
(Weisstein, Mar. 8, 2004).
The Schröder numbers bear the same relation to the Delannoy numbers as the Catalan numbers do to the binomial coefficients.
Amazingly, taking the Cholesky decomposition of the square array of 
, transposing, and multiplying it
by the diagonal matrix 
gives the square matrix (i.e., lower triangular)
version of Pascal's triangle (G. Helms,
pers. comm., Aug. 29, 2005).
Beautiful fractal patterns can be obtained by plotting 
(mod 
) (E. Pegg,
Jr., pers. comm., Aug. 29, 2005). In particular, the 
case corresponds
to a pattern resembling the Sierpiński carpet.
Banderier, C. and Schwer, S. "Why Delannoy Numbers?" To appear in J. Stat. Planning Inference. https://www-lipn.univ-paris13.fr/~banderier/Papers/delannoy2004.ps.
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 80-81, 1974.
Dickau, R. M. "Delannoy and Motzkin Numbers." https://www.prairienet.org/~pops/delannoy.html.
Goodman, E. and Narayana, T. V. "Lattice Paths with Diagonal Steps." Canad. Math. Bull. 12, 847-855, 1969.
Moser, L. "King Paths on a Chessboard." Math. Gaz. 39, 54, 1955.
Moser, L. and Zayachkowski, H. S. "Lattice Paths with Diagonal Steps." Scripta Math. 26, 223-229, 1963.
Sloane, N. J. A. Sequences A001850/M2942, A008288 , A092830, A114470, and A114491 in "The On-Line Encyclopedia of Integer Sequences."
Stocks, D. R. Jr. "Lattice Paths in 
with Diagonal
Steps." Canad. Math. Bull. 10, 653-658, 1967.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, 1991.
Weisstein, Eric W. "Delannoy Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DelannoyNumber.html
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