Catalan number

$C_n=\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=\frac{(2n)!}{(n+1)!n!}(n\ge 0)$ $C_n=\left(\genfrac{}{}{0ex}{}{2n}{n}\right)-\left(\genfrac{}{}{0ex}{}{2n}{n-1}\right)(n\ge 1)$ 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
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    • Catalan number - Wikipedia
• Length 2N, balanced Number of bracketed strings
• 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.
• With n branches (n + 1 leaves) Number of binary trees

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