# Central binomial coefficient

In mathematics the nth central binomial coefficient is the particular binomial coefficient Pascal's triangle, rows 0 through 7. The numbers in the central column are the central binomial coefficients.
${2n \choose n}={\frac {(2n)!}{(n!)^{2}}}{\text{ for all }}n\geq 0.$ 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)

## Properties

The central binomial coefficients satisfy the recurrence

${\binom {2(n+1)}{n+1}}={\frac {4n+2}{n+1}}\cdot {\binom {2n}{n}}.$

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

${\binom {2n}{n}}=(-1)^{n}4^{n}{\binom {-1/2}{n}}$

Together with the binomial series we obtain the generating function

${\frac {1}{\sqrt {1-4x}}}=\sum _{n=0}^{\infty }{\binom {2n}{n}}x^{n}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+\cdots$

and exponential generating function

$\sum _{n=0}^{\infty }{\binom {2n}{n}}{\frac {x^{n}}{n!}}=e^{2x}I_{0}(2x),$

where I0 is a modified Bessel function of the first kind.

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

${2n \choose n}\sim {\frac {4^{n}}{\sqrt {\pi n}}}.$

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 ${\sqrt {2\pi }}$  in front of the Stirling formula, by comparison.

Simple bounds that immediately follow from $4^{n}=(1+1)^{2n}=\sum _{k=0}^{2n}{\binom {2n}{k}}$  are

${\frac {4^{n}}{2n+1}}\leq {2n \choose n}\leq 4^{n}{\text{ for all }}n\geq 1$

Some better bounds are

${\frac {4^{n}}{\sqrt {\pi (n+{\frac {1}{2}})}}}\leq {2n \choose n}\leq {\frac {4^{n}}{\sqrt {\pi n}}}{\text{ for all }}n\geq 1$

and, if more accuracy is required,

${2n \choose n}={\frac {4^{n}}{\sqrt {\pi n}}}\left(1-{\frac {c_{n}}{n}}\right){\text{ where }}{\frac {1}{9}}  for all $n\geq 1.$ [citation needed]

The only central binomial coefficient that is odd is 1. More specifically, the number of factors of 2 in ${\binom {2n}{n}}$  is equal to the number of ones in the binary representation of n. Half the central binomial coefficient ${\frac {1}{2}}{2n \choose n}={2n-1 \choose n-1}$  (for $n>0$ ) (sequence A001700 in the OEIS) is seen in Wolstenholme's theorem.

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

The central binomial coefficient ${2n \choose n}$  equals the sum of the squares of the elements in row n of Pascal's triangle.

## Related sequences

The closely related Catalan numbers Cn are given by:

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

A slight generalization of central binomial coefficients is to take them as ${\frac {\Gamma (2n+1)}{\Gamma (n+1)^{2}}}={\frac {1}{n\mathrm {B} (n+1,n)}}$ , with appropriate real numbers n, where $\Gamma (x)$  is the gamma function and $\mathrm {B} (x,y)$  is the beta function.

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