Motzkin Number
The Motzkin numbers enumerate various combinatorial objects. Donaghey and Shapiro (1977) give 14 different manifestations ofthese numbers. In particular, they give the number of paths from (0, 0) to (
, 0) which never dip below 
and are madeup only of the steps (1, 0), (1, 1), and (1, 
), i.e., 
, 
, and 
. The first are 1, 2, 4,9, 21, 51, ... (Sloane's A001006). The Motzkin number Generating Function 
satisfies
 | (1) |
 | (2) |
 | (3) |
. The Motzkin number 
is also given by |  |  | (4) |
|  | ||
| (5) |
where
is a Binomial Coefficient.See also Catalan Number, King Walk, Schröder Number
References
Barcucci, E.; Pinzani, R.; and Sprugnoli, R. ``The Motzkin Family.'' Pure Math. Appl. Ser. A 2, 249-279, 1991. Donaghey, R. ``Restricted Plane Tree Representations of Four Motzkin-Catalan Equations.'' J. Combin. Th. Ser. B 22, 114-121, 1977. Donaghey, R. and Shapiro, L. W. ``Motzkin Numbers.'' J. Combin. Th. Ser. A 23, 291-301, 1977. Kuznetsov, A.; Pak, I.; and Postnikov, A. ``Trees Associated with the Motzkin Numbers.'' J. Combin. Th. Ser. A 76, 145-147, 1996. Motzkin, T. ``Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula for Partitions of a Polygon, for Permanent Preponderance, and for Nonassociative Products.'' Bull. Amer. Math. Soc. 54, 352-360, 1948. Sloane, N. J. A. SequenceA001006/M1184in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
- Concentric
- Concentric Circles
- Conchoid
- Conchoid of de Sluze
- Conchoid of Nicomedes
- Concho-Spiral
- Concordant Form
- Concur
- Concurrent
- Concyclic
- Condition
- Conditional Convergence
- Conditional Probability
- Condition Number
- Condom Problem
- Condon-Shortley Phase
- Conductor
- Cone
- Cone Graph
- Cone Net
- Cone (Space)
- Cone-Sphere Intersection
- Confidence Interval
- Configuration
- Confluent Hypergeometric Differential Equation