# central binomial coefficient

The $n$th central binomial coefficient^{} is defined to be

$$\left(\genfrac{}{}{0pt}{}{2n}{n}\right)=\frac{(2n)!}{{(n!)}^{2}}$$ |

where $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ is a binomial coefficient^{}. These numbers have the generating function^{}

$$\frac{1}{\sqrt{1-4x}}=1+2x+6{x}^{2}+20{x}^{3}+70{x}^{4}+252{x}^{5}+\mathrm{\cdots}$$ |

They are closely related to the Catalan sequence, in that

$${C}_{n}=\frac{1}{n+1}\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$$ |

Alternate definition

A less frequently-encountered definition for the $n$th central binomial coefficient is $\left(\genfrac{}{}{0pt}{}{n}{\lfloor \frac{n}{2}\rfloor}\right)$.

Note that the set of these numbers meeting this alternate criterion is a superset^{} of those meeting the first criterion, since for $n=2m$ we have

$$\left(\genfrac{}{}{0pt}{}{n}{\lfloor \frac{n}{2}\rfloor}\right)=\left(\genfrac{}{}{0pt}{}{2m}{\lfloor \frac{2m}{2}\rfloor}\right)=\left(\genfrac{}{}{0pt}{}{2m}{m}\right)$$ |

By cancelling terms of one of the $n!$’s against terms of the $2n!$, one may rewrite the central binomial coefficient as follows:

$$\left(\genfrac{}{}{0pt}{}{2n}{n}\right)=\frac{2n(2n-1)\mathrm{\cdots}(n+2)(n+1)}{n(n-1)\mathrm{\cdots}3\cdot 2\cdot 1}.$$ |

Alternatively, one may cancel each term of the $n!$ against twice itself, leaving $2$’s in the numerator:

$$\left(\genfrac{}{}{0pt}{}{2n}{n}\right)={2}^{n}\frac{(2n-1)(2n-3)\mathrm{\cdots}5\cdot 3\cdot 1}{n(n-1)\mathrm{\cdots}3\cdot 2\cdot 1}$$ |

Doubling the terms in the denominator, we obtain an expression for the central binomial coeficient
in terms of a quotient of successive odd numbers^{} by successive even numbers:

$$\left(\genfrac{}{}{0pt}{}{2n}{n}\right)={4}^{n}\frac{(2n-1)(2n-3)\mathrm{\cdots}5\cdot 3\cdot 1}{2n(2n-2)\mathrm{\cdots}6\cdot 4\cdot 2}$$ |

By means of these formulae, one may derive some important properties of the central
binomial coeficients. By examining the first two formulae, one may deduce results
about the prime factors^{} of central binomial coefficients (for proofs, please see the
attachments to this entry):

###### Theorem 1

If $n\mathrm{\ge}\mathrm{3}$ is an integer and $p$ is a prime number^{} such that $$, then
$p$ divides $\mathrm{\left(}\genfrac{}{}{0pt}{}{\mathrm{2}\mathit{}n}{n}\mathrm{\right)}$.

###### Theorem 2

If $n\mathrm{\ge}\mathrm{3}$ is an integer and $p$ is a prime number such that $$, then $p$ does not divide $\mathrm{\left(}\genfrac{}{}{0pt}{}{\mathrm{2}\mathit{}n}{n}\mathrm{\right)}$.

In conjunction^{} with Wallis’ formula^{} for $\pi $, the third formula for the central
binomial coefficient may be used to derive an asymptotic expression, as is done in
an attachment to this entry:

$$\left(\genfrac{}{}{0pt}{}{2n}{n}\right)\approx \sqrt{\frac{2}{\pi}}\frac{{4}^{n}}{\sqrt{2n+1}}$$ |

Title | central binomial coefficient |
---|---|

Canonical name | CentralBinomialCoefficient |

Date of creation | 2013-03-22 14:25:40 |

Last modified on | 2013-03-22 14:25:40 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 05A10 |

Classification | msc 11B65 |

Related topic | BinomialCoefficient |

Related topic | CatalanNumbers |