Catalan sequence

The Catalan sequence is a sequence of positive integers that arise as the solution to a wide variety of combinatorial problems. The first few terms of the Catalan sequence are $C_0 = 1$, $C_1 = 1$, $C_2 = 2$, $C_3 = 5$, .... In general, the $n$th term of the Catalan sequence is given by the formula $C_n = \frac{1}{n + 1}\binom{2n}{n}$, where $\binom{2n}{n}$ is the $n$th central binomial coefficient.


The Catalan sequence can be used to find:

  1. The number of ways to arrange $n$ pairs of matching parentheses.
  2. The number of ways a convex polygon of $n+2$ sides can be split into $n$ triangles by $n - 1$ nonintersection diagonals.
  3. The number of rooted binary trees with exactly $n+1$ leaves.
  4. The number of paths with $2n$ steps on a rectangular grid from $(0,0)$ to $(n,n)$ that do not cross above the main diagonal.

A recursive definition of the Catalan sequence is $C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k}$


In how many ways can the product of $n$ ordered number be calculated by pairs? For example, the possible ways for $a\cdot b\cdot c\cdot d$ are $a((bc)d), a(b(cd)), (ab)(cd), ((ab)c)d,$ and $(a(bc))d$.


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