Delannoy Number
المؤلف:
Banderier, C. and Schwer, S
المصدر:
"Why Delannoy Numbers?" To appear in J. Stat. Planning Inference. http://www-lipn.univ-paris13.fr/~banderier/Papers/delannoy2004.ps.
الجزء والصفحة:
...
16-9-2021
1213
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
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
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
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.
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 
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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