View source for Central binomial coefficient

{{Use American English|date = March 2019}}
{{Short description|Sequence of numbers ((2n) choose (n))}}
{{no footnotes|date=June 2016}}
[[File:Pascal triangle small.png|thumb|right|300px|Pascal's triangle, rows 0 through 7. The numbers in the central column are the central binomial coefficients.]]
In [[mathematics]] the ''n''th '''central binomial coefficient''' is the particular [[binomial coefficient]]

: <math>{2n \choose n} = \frac{(2n)!}{(n!)^2}\text{ for all }n \geq 0.</math>

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:

:{{num|1}}, {{num|2}}, {{num|6}}, {{num|20}}, {{num|70}}, {{num|252}}, 924, 3432, 12870, 48620, ...; {{OEIS|id=A000984}}

== Properties ==
The central binomial coefficients satisfy the recurrence

:<math>\binom{2(n+1)}{n + 1} = \frac{4n + 2}{n + 1} \cdot \binom{2n}{n}.</math>

Since <math>\textstyle\binom{-1/2}{n + 1} = \frac{-1/2 - n}{n + 1} \cdot \binom{-1/2}{n}</math> we find 

:<math>\binom{2n}{n} = (-1)^n 4^n \binom{-1/2}{n} </math> 

Together with the [[binomial series]] we obtain the [[generating function]]

:<math>\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</math>

and exponential generating function

:<math> \sum_{n=0}^\infty \binom{2n}{n}\frac{x^n}{n!} = e^{2x} I_0(2x), </math>
where ''I''<sub>0</sub> is a [[Bessel function|modified Bessel function of the first kind]].<ref name=Sloanes>{{Cite OEIS|sequencenumber=A000984|name=Central binomial coefficients}}</ref>

The [[Wallis product]] can be written in asymptotic form for the central binomial coefficient:

: <math> {2n \choose n} \sim \frac{4^n}{\sqrt{\pi n}}.</math>

The latter can also be easily established by means of [[Stirling's formula]]. On the other hand, it can also be used as a means to determine the constant <math>\sqrt{2\pi}</math> in front of the Stirling formula, by comparison.

Simple bounds that immediately follow from <math>4^n=(1+1)^{2n}= \sum_{k=0}^{2n} \binom{2n}{k}</math> are

:<math>\frac{4^n}{2n+1} \leq {2n \choose n} \leq 4^n\text{ for all }n \geq 1</math>

Some better bounds are<ref>Kazarinoff, N.D. ''Geometric inequalities'', New York: Random House, 1961</ref>

:<math>\frac{4^n}{\sqrt{4n}} \leq {2n \choose n} \leq \frac{4^n}{\sqrt{3n+1}}\text{ for all }n \geq 1</math>

and, if more accuracy is required,

:<math>{2n \choose n} = \frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right)\text{ where }\frac{1}{9} < c_n < \frac{1}{8}</math> for all <math>n \geq 1.</math>
{{citation needed|reason=There should be a book to refer to|date=March 2016}}

The only central binomial coefficient that is odd is 1.  More specifically, the number of factors of 2 in <math>\binom{2n}{n}</math> is equal to the number of ones in the [[binary number|binary]] representation of ''n''.<ref>{{Cite OEIS|sequencenumber=A000120}}</ref>

By the [[Erdős squarefree conjecture]], proven in 1996, no central binomial coefficient with ''n'' > 4 is [[squarefree integer|squarefree]].

The central binomial coefficient <math>{2n \choose n}</math> equals the sum of the squares of the elements in row ''n'' of Pascal's triangle.<ref name=Sloanes/>

==Related sequences==
The closely related [[Catalan numbers]] ''C''<sub>''n''</sub> are given by:

:<math>C_n = \frac{1}{n+1} {2n \choose n} = {2n \choose n} -
        {2n \choose n+1}\text{ for all }n \geq 0.</math>

A slight generalization of central binomial coefficients is to take them as
<math> \frac{\Gamma(2n+1)}{\Gamma(n+1)^2}=\frac{1}{n \Beta(n+1,n)}</math>, with appropriate real numbers ''n'', where <math>\Gamma(x)</math> is the [[gamma function]] and <math>\Beta(x,y)</math> is the [[beta function]].

The [[power of two|powers of two]] that divide the central binomial coefficients are given by [[Gould's sequence]], whose ''n''th element is the number of odd integers in row ''n'' of Pascal's triangle.

==References==
{{Reflist}}
* {{citation
 | last = Koshy
 | first = Thomas 
 | title = Catalan Numbers with Applications
 | publisher = Oxford University Press
 | year = 2008
 | isbn = 978-0-19533-454-8
}}.

==External links==
* {{planetmath|urlname=centralbinomialcoefficient|title=Central binomial coefficient}}
* {{planetmath|urlname=binomialcoefficient|title=Binomial coefficient}}
* {{planetmath|urlname=pascalstriangle|title=Pascal's triangle}}
* {{planetmath|urlname=catalannumbers|title=Catalan numbers}}

{{PlanetMath attribution|urlname=centralbinomialcoefficient|title=Central binomial coefficient}}

[[Category:Factorial and binomial topics]]