Central Binomial Coefficient

In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient

: $= \frac = \prod\limits_^\frac \textn \geq 0.$

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:

:, , , , , , 924, 3432, 12870, 48620, ...;

Properties

The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects.

For example, $n=2$ represents ''AABB, ABAB, ABBA, BAAB, BABA, BBAA''.

They also represent the number of combinations of ''A'' and ''B'' where there are never more ''B'' 's than ''A'' 's.

For example, $n=2$ represents ''AAAA, AAAB, AABA, AABB, ABAA, ABAB''.

The number of factors of ''2'' in $\binom$ is equal to the number of ones in the binary representation of ''n'', so ''1'' is the only odd central binomial coefficient.

Generating function

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

The central binomial coefficients have exponential generating function
$\sum_^\infty \binom\frac = e^ I_0(2x),$
where ''I''₀ is a modified Bessel function of the first kind.

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:
:$\sum_^ \binom^2 x^ = \frac K\left(\frac\right).$

Asymptotic growth

The Wallis product can be written using limits:

:$\frac = \lim_ \prod_^n \frac = \lim_ \frac = \lim_ 4^nn!^2\frac$

because $(2n)!=2^nn!(2n-1)!!$.

Taking the square root of both sides gives the asymptote for the central binomial coefficient:

:$\sim \frac$.

The latter can also be established by means of Stirling's formula. On the other hand, it can also be used as a means to determine the constant $\sqrt$ in front of the Stirling formula.

Approximations

Simple bounds that immediately follow from $4^n=(1+1)^= \sum_^ \binom$ are

:$\frac \leq \leq 4^n\textn \geq 0$

Some better bounds are

:$\frac \leq \leq \frac\textn \geq 1$

Related sequences

The closely related Catalan numbers ''C''_''n'' are given by:

:$C_n = \frac = - \textn \geq 0.$

A slight generalization of central binomial coefficients is to take them as
$\frac=\frac$, with appropriate real numbers ''n'', where $\Gamma(x)$ is the gamma function and $\Beta(x,y)$ 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

:$\frac = \sum_^\infty \binom x^n \sum_^\infty \binom x^n$

Comparing the coefficients of $x^n$ gives

:$\sum_^n \binom\binom = 4^n$

For example, $64=1(20)+2(6)+6(2)+20(1)$.

Other information

Half the central binomial coefficient $\textstyle\frac12 =$ (for $n>0$) is seen in Wolstenholme's theorem.

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

$\textstyle \binom$ is the sum of the squares of the ''n''-th row of Pascal's Triangle:

:$=\sum_^n \binom^2$

For example, $\tbinom=20=1^2+3^2+3^2+1^2$.

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

Another noteworthy fact is that the power of 2 dividing $(n+1)\dots(2n)$ is exactly .

See also

* Central trinomial coefficient

References

* .

External links

* {{planetmath, urlname=centralbinomialcoefficient, title=Central binomial coefficient

Factorial and binomial topics