In mathematics the nth central binomial coefficient is the particular binomial coefficient

${\displaystyle {2n \choose n}={\frac {(2n)!}{(n!)^{2}}}=\prod \limits _{k=1}^{n}{\frac {n+k}{k}}{\text{ for all }}n\geq 0.}$

They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at n = 0 are:

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...; (sequence A000984 in the OEIS)

Combinatorial interpretations and other properties

The central binomial coefficient ${\displaystyle {\binom {2n}{n}}}$ is the number of arrangements where there are an equal number of two types of objects. For example, when ${\displaystyle n=2}$, the binomial coefficient ${\displaystyle {\binom {2\cdot 2}{2}}}$ is equal to 6, and there are six arrangements of two copies of A and two copies of B: AABB, ABAB, ABBA, BAAB, BABA, BBAA.

The same central binomial coefficient ${\displaystyle {\binom {2n}{n}}}$ is also the number of words of length 2n made up of A and B within which, as one reads from left to right, there are never more B's than A's at any point. For example, when ${\displaystyle n=2}$, there are six words of length 4 in which each prefix has at least as many copies of A as of B: AAAA, AAAB, AABA, AABB, ABAA, ABAB.

The number of factors of 2 in ${\displaystyle {\binom {2n}{n}}}$ is equal to the number of 1s in the binary representation of n.^[1] As a consequence, 1 is the only odd central binomial coefficient.

Generating function

The ordinary generating function for the central binomial coefficients is $${\displaystyle {\frac {1}{\sqrt {1-4x}}}=\sum _{n=0}^{\infty }{\binom {2n}{n}}x^{n}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+\cdots .}$$ This can be proved using the binomial series and the relation $${\displaystyle {\binom {2n}{n}}=(-1)^{n}4^{n}{\binom {-1/2}{n}},}$$ where ${\displaystyle \textstyle {\binom {-1/2}{n}}}$ is a generalized binomial coefficient.^[2]

The central binomial coefficients have exponential generating function $${\displaystyle \sum _{n=0}^{\infty }{\binom {2n}{n}}{\frac {x^{n}}{n!}}=e^{2x}I_{0}(2x),}$$ where I₀ is a modified Bessel function of the first kind.^[3]

The generating function of the squares of the central binomial coefficients can be written in terms of the complete elliptic integral of the first kind:^[4]

${\displaystyle \sum _{n=0}^{\infty }{\binom {2n}{n}}^{2}x^{n}={\frac {2}{\pi }}K(4{\sqrt {x}}).}$

Asymptotic growth

The asymptotic behavior can be described quite accurately:^[5] $${\displaystyle {2n \choose n}={\frac {4^{n}}{\sqrt {\pi n}}}\left(1-{\frac {1}{8n}}+{\frac {1}{128n^{2}}}+{\frac {5}{1024n^{3}}}+O(n^{-4})\right).}$$

Related sequences

The closely related Catalan numbers C_n are given by:

${\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={2n \choose n}-{2n \choose n+1}{\text{ for all }}n\geq 0.}$

A slight generalization of central binomial coefficients is to take them as ${\displaystyle \frac{\mathrm{\Gamma <!--\; \Gamma \; -->}(2n+1)}{\mathrm{\Gamma <!--\; \Gamma \; -->}(n+1{)}^2}}$Zdroj:https://en.wikipedia.org?pojem=Central_binomial_coefficient
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