In mathematics, a Delannoy number describes the number of 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.
The Delannoy number also counts the number of global alignments of two sequences of lengths and , the number of points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the number of cells in an m-dimensional von Neumann neighborhood of radius n while the number of cells on a surface of an m-dimensional von Neumann neighborhood of radius n is given with .
Example
The Delannoy number D(3,3) equals 63. The following figure illustrates the 63 Delannoy paths from (0, 0) to (3, 3):
The subset of paths that do not rise above the SW–NE diagonal are counted by a related family of numbers, the Schröder numbers.
Delannoy array
The Delannoy array is an infinite matrix of the Delannoy numbers:
{ | class="wikitable" style="text-align:right;" |- ! ! width="50" | 0 ! width="50" | 1 ! width="50" | 2 ! width="50" | 3 ! width="50" | 4 ! width="50" | 5 ! width="50" | 6 ! width="50" | 7 ! width="50" | 8 |- ! 0 | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |- ! 1 | 1 || 3 || 5 || 7 || 9 || 11 || 13 || 15 || 17 |- ! 2 | 1 || 5 || 13 || 25 || 41 || 61 || 85 || 113 || 145 |- ! 3 | 1 || 7 || 25 || 63 || 129 || 231 || 377 || 575 || 833 |- ! 4 | 1 || 9 || 41 || 129 || 321 || 681 || 1289 || 2241 || 3649 |- ! 5 | 1 || 11 || 61 || 231 || 681 || 1683 || 3653 || 7183 || 13073 |- ! 6 | 1 || 13 || 85 || 377 || 1289 || 3653 || 8989 || 19825 || 40081 |- ! 7 | 1 || 15 || 113 || 575 || 2241 || 7183 || 19825 || 48639 || 108545 |- ! 8 | 1 || 17 || 145 || 833 || 3649 || 13073 || 40081 || 108545 || 265729 |- ! 9 | 1 || 19 || 181 || 1159 || 5641 || 22363 || 75517 || 224143 || 598417 |}
In this array, the numbers in the first row are all one, the numbers in the second row are the odd numbers, the numbers in the third row are the centered square numbers, and the numbers in the fourth row are the centered octahedral numbers. Alternatively, the same numbers can be arranged in a triangular array resembling Pascal's triangle, also called the tribonacci triangle, in which each number is the sum of the three numbers above it:
1
1 1
1 3 1
1 5 5 1
1 7 13 7 1
1 9 25 25 9 1
1 11 41 63 41 11 1
Central Delannoy numbers
The central Delannoy numbers D(n) = D(n,n) are the numbers for a square n × n grid. The first few central Delannoy numbers (starting with n=0) are:
1, 3, 13, 63, 321, 1683, 8989, 48639, 265729, ... .
Computation
Delannoy numbers
For diagonal (i.e. northeast) steps, there must be steps in the direction and steps in the direction in order to reach the point ; as these steps can be performed in any order, the number of such paths is given by the multinomial coefficient . Hence, one gets the closed-form expression
An alternative expression is given by
or by the infinite series
And also
where is given with .
The basic recurrence relation for the Delannoy numbers is easily seen to be
This recurrence relation also leads directly to the generating function
Central Delannoy numbers
Substituting in the first closed form expression above, replacing , and a little algebra, gives
while the second expression above yields
The central Delannoy numbers satisfy also a three-term recurrence relationship among themselves,
and have a generating function
The leading asymptotic behavior of the central Delannoy numbers is given by
where
and
.