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The Catalan Numbers express the number of ways you can divide a polygon with N sides into triangles, using nonintersecting diagonals.
no. of polygon sides 3
4
5
6
7
8
9
no. of ways to divide 1
2
5
14
42
132
429
These numbers are today called the Catalan numbers after the Belgian Eugène Charles Catalan, who however was not the first to solve the problem, since Segner had solved it in the 18th century and Euler and Binet also worked on simplifying the solution.
Seth Johnson writes that in 1838, Catalan "solved the problem of how many ways one can parenthesize a chain of N+1 letters using N pairs of parentheses such that there are either two letters, a parenthesized expression and a letter, or two parenthesized expressions within each pair of parentheses. The answer for any positive number N is the Nth Catalan Number."
The Catalan Numbers can be computed using this formula:
and there are at least two ways they can be found in Pascal's triangle, one in the middle column going down the center, subtracting the element immediately adjacent (see numbers in red), and another one row above, taking the Nth term over and subtracting the term immediately to the right (numbers in green).








1 















1 

1 













1 

2 
 
1 











1 

3 

3 
 
1 









1 

4 

6 
 
4 

1 







1 

5 

10 

10 
 
5 

1 





1 

6 

15 

20 
 
15 

6 

1 



1 

7 

21 

35 

35 
 
21 

7 

1 

1 

8 

28 

56 

70 
 
56 

28 

8 

1 





... 

126 

126 
 
84 
... 









... 
210 

252 
 
210 

... 









... 

462 

462 
 
330 
... 









... 
792 

924 
 
792 

... 




References
 Robert Dickau: Catalan Numbers
 Brian Hayes: A Question of Numbers
 Seth Johnson: The Catalan Numbers in Pascal's Triangle
 Kevin Brown: The Meanings of Catalan Numbers
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