Central binomial coefficient

Definition

Let $n$ be a natural number. The $n$-th central binomial coefficient is the binomial coefficient $$\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.$$

Remarks

The first values, for $n = 0, 1, 2, 3, 4$, are $1, 2, 6, 20, 70$. It is the middle entry of row $2n$ of Pascal's triangle, and the largest entry of that row. It counts the lattice paths from $(0, 0)$ to $(n, n)$ with unit steps right and up. The closed formula for Catalan numbers divides it by $n + 1$.

Explanations

L1 human

The central binomial coefficient is the number in the middle of an even row of Pascal's triangle. For $n = 2$ it is $\binom{4}{2} = 6$: the ways to choose 2 of 4 objects.