Catalan number

$ C_n ={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}\quad (n\geq 0)

$ C_{n}={2n \choose n}-{2n \choose n-1}\quad (n\geq 1)

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, …

Number of cases in which the operation is repeated to move forward by 1 in the direction of increasing x or y coordinates, from (0, 0) to (N, N) without passing through the region where x < y

https://gyazo.com/091f70ce1d43c2c68dc6fecc9593c68e

Number of triangulations] of convex n-angles

Consider a convex polygon consisting of n + 2 edges, divided into n triangles by drawing lines connecting the vertices so that they do not intersect each other. The number of cases for this division is the Catalan number Cn.

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