Catalan numbers (original) (raw)
The nth Catalan number is given by:
where (nr) represents the binomial coefficient
. The first several Catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862 ,…(see OEIS sequence
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000108A000108 for more terms). The Catalan numbers are also generated by the recurrence relation
| C0=1,Cn=∑i=0n-1CiCn-1-i. |
|---|
For example, C3=1⋅2+1⋅1+2⋅1=5, C4=1⋅5+1⋅2+2⋅1+5⋅1=14, etc.
The ordinary generating function for the Catalan numbers is
Interpretations of the nth Catalan number include:
- The number of ways to arrange n pairs of matching parentheses, e.g.:
((())) (()()) ()(()) (())() ()()()
- The number of ways to arrange n pairs of matching parentheses, e.g.:
- The number of ways a convex polygon of n+2 sides can be split into n triangles
.
- The number of ways a convex polygon of n+2 sides can be split into n triangles
- The number of rooted binary trees with exactly n+1 leaves.
The Catalan sequence is named for Eugène Charles Catalan, but it was discovered in 1751 by Euler when he was trying to solve the problem of subdividing polygons into triangles.
References
- 1 Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, 1998. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0836.00001Zbl 0836.00001.