{{Use American English|date = March 2019}} {{Short description|Sequence of numbers ((2n) choose (n))}} 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}}

==Combinatorial interpretations and other properties== thumb|The central binomial coefficients give the number of possible number of assignments of ''n''-a-side sports teams from 2''n'' players, taking into account the playing area side The central binomial coefficient <math>\binom{2n}{n}</math> is the number of arrangements where there are an equal number of two types of objects. For example, when <math>n=2</math>, the binomial coefficient <math>\binom{2 \cdot 2}{2}</math> 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 <math>\binom{2n}{n}</math> is also the number of words of length 2''n'' 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 <math>n=2</math>, 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 <math>\binom{2n}{n}</math> is equal to the number of 1s in the binary representation of ''n''.<ref>{{Cite OEIS|sequencenumber=A000120}}</ref> As a consequence, 1 is the only odd central binomial coefficient.

==Generating function==

The ordinary generating function for the central binomial coefficients is <math display="block">\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> This can be proved using the binomial series and the relation <math display="block">\binom{2n}{n} = (-1)^n 4^n \binom{-1/2}{n}, </math> where <math>\textstyle\binom{-1/2}{n}</math> is a generalized binomial coefficient.<ref>{{citation | last = Stanley | first = Richard P. | authorlink = Richard P. Stanley | title = Enumerative Combinatorics | volume = 1 | edition = 2 | at = Example 1.1.15 | publisher = Cambridge University Press | year = 2012 | isbn = 978-1-107-60262-5}}</ref>

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

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:<ref>{{cite OEIS|A002894}}</ref> :<math>\sum_{n=0}^{\infty} \binom{2n}{n}^2 x^{n} = \frac{2}{\pi}K(4\sqrt{x}).</math>

==Asymptotic growth== The asymptotic behavior can be described quite accurately:<ref>{{cite book |last1=Luke |first1=Yudell L. |authorlink = Yudell Luke |title=The Special Functions and their Approximations, Vol. 1 |date=1969 |publisher=Academic Press, Inc. |location=New York, NY, USA |page=35}}</ref> <math display="block"> {2n \choose n} = \frac{4^n}{\sqrt{\pi n}}\left(1-\frac{1}{8n}+\frac{1}{128 n^2}+\frac{5}{1024 n^3} + O(n^{-4})\right).</math>

==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 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.

Squaring the generating function gives <math display="block"> \frac{1}{1-4x} = \left(\sum_{n=0}^\infty \binom{2n}{n} x^n\right)\left(\sum_{n=0}^\infty \binom{2n}{n} x^n\right). </math> Comparing the coefficients of <math>x^n</math> gives <math display="block"> \sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k} = 4^n. </math> For example, <math>64=1(20)+2(6)+6(2)+20(1)</math> {{OEIS|A000302}}.

The number lattice paths of length 2''n'' that consist of steps of unit length in one of the four cardinal directions and start and end at the origin is <math display="block"> \sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k}\binom{2n}{2k} = \binom{2n}{n}\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}^2 </math> {{OEIS|A002894}}.

==Properties==

Half the central binomial coefficient <math>\textstyle\frac12{2n \choose n} = {2n-1 \choose n-1}</math> (for <math>n>0</math>) {{OEIS|id=A001700}} is seen in Wolstenholme's theorem.

By the Erdős squarefree conjecture, proved in 1996, no central binomial coefficient with ''n'' > 4 is squarefree.

<math>\textstyle \binom{2n}{n}</math> is the sum of the squares of the ''n''-th row of Pascal's Triangle:<ref name=Sloanes/>

:<math>{2n \choose n}=\sum_{k=0}^n \binom{n}{k}^2</math>

For example, <math>\tbinom{6}{3}=20=1^2+3^2+3^2+1^2</math>.

Erdős uses central binomial coefficients extensively in his proof of Bertrand's postulate.

Another noteworthy fact is that the power of 2 dividing <math>(n+1)\dots(2n)</math> is exactly {{mvar|n}}.

== See also == * Central trinomial coefficient

==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}}

Category:Factorial and binomial topics