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Delannoy Number


The Delannoy numbers D(a,b) are the number of lattice paths from (0,0) to (b,a) 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

 D(a,b)=D(a-1,b)+D(a,b-1)+D(a-1,b-1),
(1)

with D(0,0)=1. The are also given by the sums

D(n,k)=sum_(d=0)^(n)(k; d)(n+k-d; k)
(2)
=sum_(d=0)^(n)2^d(k; d)(n; d)
(3)
=(n+k; k)_2F_1(-n,-k;-(k+n);-1),
(4)

where _2F_1(a,b;c;z) is a hypergeometric function.

A table for values for the Delannoy numbers is given by

 1 1 1 1 1 1 1 1 1 ...; 1 3 5 7 9 11 13 15 17 ...; 1 5 13 25 41 61 85 113 145 ...; 1 7 25 63 129 231 377 575 833 ...; 1 9 41 129 321 681 1289 2241 3649 ...; 1 11 61 231 681 1683 3653 7183 13073 ...
(5)

(OEIS A008288) for m=0, 1, ... increasing from left to right and n=0, 1, ... increasing from top to bottom.

They have the generating function

 sum_(p,q=1)^inftyD(p,q)x^py^q=(1-x-y-xy)^(-1)
(6)

(Comtet 1974, p. 81).

DelannoyNumber

Taking n=a=b gives the central Delannoy numbers D(n,n), which are the number of "king walks" from the (0,0) corner of an n×n square to the upper right corner (n,n). These are given by

 D(n,n)=P_n(3),
(7)

where P_n(x) is a Legendre polynomial (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression is

D(n)=D(n,n)
(8)
=sum_(k=0)^(n)(n; k)(n+k; k)
(9)
=_2F_1(-n,n+1;1,-1),
(10)

where (a; b) is a binomial coefficient and _2F_1(a,b;c;z) 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

 D(n)=(3(2n-1)D(n-1)-(n-1)D(n-2))/n.
(11)

They have generating function

G(x)=1/(sqrt(1-6x+x^2))
(12)
=1+3x+13x^2+63x^3+321x^4+....
(13)

The values of D(n) for n=1, 2, ... are 3, 13, 63, 321, 1683, 8989, 48639, ... (OEIS A001850). The numbers of decimal digits in D(10^n,10^n) for n=0, 1, ... are 1, 7, 76, 764, 7654, 76553, 765549, 7655510, ... (OEIS A114470), where the digits approach those of log_(10)(3+2sqrt(2))=0.765551... (OEIS A114491).

The first few prime Delannoy numbers are 3, 13, 265729, ... (OEIS A092830), corresponding to indices 1, 2, 8, ..., with no others for n<1.1×10^5 (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 D(a,b), transposing, and multiplying it by the diagonal matrix diag(2^(-0/2),2^(-1/2),2^(-2/2),...) gives the square matrix (i.e., lower triangular) version of Pascal's triangle (G. Helms, pers. comm., Aug. 29, 2005).

DelannoyNumberArrays

Beautiful fractal patterns can be obtained by plotting D(a,b) (mod m) (E. Pegg, Jr., pers. comm., Aug. 29, 2005). In particular, the m=3 case corresponds to a pattern resembling the Sierpiński carpet.


See also

Binomial Coefficient, Cantor Function, Catalan Number, Integer Sequence Primes, Motzkin Number, Schmidt's Problem, Schröder Number

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References

Banderier, C. and Schwer, S. "Why Delannoy Numbers?" To appear in J. Stat. Planning Inference. http://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." http://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 E^3 with Diagonal Steps." Canad. Math. Bull. 10, 653-658, 1967.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, 1991.

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Delannoy Number

Cite this as:

Weisstein, Eric W. "Delannoy Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DelannoyNumber.html

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