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.
with Diagonal Steps." Canad. Math. Bull. 10, 653-658, 1967.Vardi,
I.