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In mathematics, a Delannoy number D 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 D(m,n) also counts the number of global alignments of two sequences of lengths m and n, the number of points in an m-dimensional integer lattice that 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.

Example

The Delannoy number D(3,3) equals 63. The following figure illustrates the 63 Delannoy paths through a 3 × 3 grid: 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:

n \ m 0 1 2 3 4 5 6 7 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, ... (sequence A001850 in OEIS).

Computation
Delannoy numbers

For k diagonal (i.e. northeast) steps, there must be m-k steps in the x direction and n-k steps in the y direction in order to reach the point (m, n) ; as these steps can be performed in any order, the number of such paths is given by the multinomial coefficient $$\binom{m+n-k}{k , m-k , n-k} = \binom{m+n-k}{m} \binom{m}{k}$$ . Hence, one gets the closed-form expression

$$D(m,n) = \sum_{k=0}^{\min(m,n)} \binom{m+n-k}{m} \binom{m}{k}$$ .

An alternative expression is given by

$$D(m,n) = \sum_{k=0}^{\min(m,n)} \binom{m}{k} \binom{n}{k} 2^k$$ .

The basic recurrence relation for the Delannoy numbers is easily seen to be

$$D(m,n)=\begin{cases}1 &\text{if }m=0\text{ or }n=0\\D(m-1,n) + D(m-1,n-1) + D(m,n-1)&\text{otherwise}\end{cases}$$

This recurrence relation also leads directly to the generating function

$$\sum_{m,n = 0}^\infty D(m, n) x^m y^n = (1 - x - y - xy)^{-1}$$ .

Central Delannoy numbers

Substituting m = n in the first closed form expression above, replacing $$k \leftrightarrow n-k$$, and a little algebra, gives

$$D(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} ,$$

while the second expression above yields

$$D(n) = \sum_{k=0}^n \binom{n}{k}^2 2^k .$$

The central Delannoy numbers satisfy also a three-term recurrence relationship among themselves,

$$n D(n) = 3(2n-1)D(n-1) - (n-1)D(n-2) ,$$

and have a generating function

$$\sum_{n = 0}^\infty D(n) x^n = (1-6x+x^2)^{-1/2} .$$

The leading asymptotic behavior of the central Delannoy numbers is given by

$$D(n) = \frac{c \, \alpha^n}{\sqrt{n}} \, (1 + O(n^{-1}))$$

where $$\alpha = 3 + 2 \sqrt{2} \approx 5.828 and c = (4 \pi (3 \sqrt{2} - 4))^{-1/2} \approx 0.5727$$ .

Motzkin number
Narayana number

References

Banderier, Cyril; Schwer, Sylviane (2005), "Why Delannoy numbers?", Journal of Statistical Planning and Inference 135 (1): 40–54, arXiv:math/0411128, doi:10.1016/j.jspi.2005.02.004
Covington, Michael A. (2004), "The number of distinct alignments of two strings", Journal of Quantitative Linguistics 11 (3): 173–182, doi:10.1080/0929617042000314921
Luther, Sebastian; Mertens, Stephan (2011), "Counting lattice animals in high dimensions", Journal of Statistical Mechanics: Theory and Experiment 2011 (9): P09026, arXiv:1106.1078
Breukelaar, R.; Bäck, Th. (2005), "Using a Genetic Algorithm to Evolve Behavior in Multi Dimensional Cellular Automata: Emergence of Behavior", Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO '05), New York, NY, USA: ACM, pp. 107–114, doi:10.1145/1068009.1068024, ISBN 1-59593-010-8
Sulanke, Robert A. (2003), "Objects counted by the central Delannoy numbers" (PDF), Journal of Integer Sequences 6 (1): Article 03.1.5, MR 1971435
"Sloane's A008288 : Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Peart, Paul; Woan, Wen-Jin (2002). "A bijective proof of the Delannoy recurrence". Congressus Numerantium 158: 29–33. ISSN 0384-9864. Zbl 1030.05003.