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Summary Pascal's triangle, rows 0 through 7. The numbers in the central column are the central binomial coefficients.

In mathematics the nth central binomial coefficient is the particular binomial coefficient

${2n \choose n}={\frac {(2n)!}{(n!)^{2}}}=\prod \limits _{k=1}^{n}{\frac {n+k}{k}}{\text{ for all }}n\geq 0.$ 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 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 ${\binom {2n}{n}}$ 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

${\frac {1}{\sqrt {1-4x}}}=\sum _{n=0}^{\infty }{\binom {2n}{n}}x^{n}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+\cdots .$ This can be proved using the binomial series and the relation
${\binom {2n}{n}}=(-1)^{n}4^{n}{\binom {-1/2}{n}},$ where $\textstyle {\binom {-1/2}{n}}$ is a generalized binomial coefficient.

The central binomial coefficients have 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 generating function of the squares of the central binomial coefficients can be written in terms of the complete elliptic integral of the first kind:[citation needed]

$\sum _{n=0}^{\infty }{\binom {2n}{n}}^{2}x^{n}={\frac {4}{\pi (1+{\sqrt {1-16x}})}}K\left({\frac {1-{\sqrt {1-16x}}}{1+{\sqrt {1-16x}}}}\right).$ Asymptotic growth

The Wallis product can be written using limits:

${\frac {\pi }{2}}=\lim _{n\to \infty }\prod _{k=1}^{n}{\frac {2k\cdot 2k}{(2k-1)(2k+1)}}=\lim _{n\to \infty }{\frac {4^{n}n!^{2}}{(2n-1)!!(2n+1)!!}}=\lim _{n\to \infty }4^{n}n!^{2}{\frac {2^{2n}n!^{2}}{(2n)!^{2}(2n+1)}}$ because $(2n)!=2^{n}n!(2n-1)!!$ .

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

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

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

Approximations

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

${\frac {4^{n}}{2n+1}}\leq {2n \choose n}\leq 4^{n}{\text{ for all }}n\geq 0$ Some better bounds are

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

Squaring the generating function gives

${\frac {1}{1-4x}}=\sum _{n=0}^{\infty }{\binom {2n}{n}}x^{n}\sum _{n=0}^{\infty }{\binom {2n}{n}}x^{n}$ Comparing the coefficients of $x^{n}$ gives

$\sum _{k=0}^{n}{\binom {2k}{k}}{\binom {2n-2k}{n-k}}=4^{n}$ For example, $64=1(20)+2(6)+6(2)+20(1)$ . (sequence A000302 in the OEIS)

Other information

Half the central binomial coefficient $\textstyle {\frac {1}{2}}{2n \choose n}={2n-1 \choose n-1}$ (for $n>0$ ) (sequence A001700 in the OEIS) is seen in Wolstenholme's theorem.

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

$\textstyle {\binom {2n}{n}}$ is the sum of the squares of the n-th row of Pascal's Triangle:

${2n \choose n}=\sum _{k=0}^{n}{\binom {n}{k}}^{2}$ For example, ${\tbinom {6}{3}}=20=1^{2}+3^{2}+3^{2}+1^{2}$ .

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 n.

References

1. ^ Sloane, N. J. A. (ed.). "Sequence A000120". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Stanley, Richard P. (2012), Enumerative Combinatorics, 1 (2 ed.), Cambridge University Press, Example 1.1.15, ISBN 978-1-107-60262-5
3. ^ a b Sloane, N. J. A. (ed.). "Sequence A000984 (Central binomial coefficients)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Koshy, Thomas (2008), Catalan Numbers with Applications, Oxford University Press, ISBN 978-0-19533-454-8.