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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

(2nn)=(2n)!(n!)2 for all n≥0.{displaystyle {2n choose n}={frac {(2n)!}{(n!)^{2}}}{text{ for all }}ngeq 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 have ordinary generating function

11−4x=1+2x+6x2+20x3+70x4+252x5+⋯{displaystyle {frac {1}{sqrt {1-4x}}}=1+2x+6x^{2}+20x^{3}+70x^{4}+252x^{5}+cdots }

and exponential generating function

∑n=0∞(2nn)xnn!=e2xI0(2x),{displaystyle 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.^[1]

They also satisfy the recurrence

(2(n+1)n+1)=4n+2n+1⋅(2nn).{displaystyle {binom {2(n+1)}{n+1}}={frac {4n+2}{n+1}}cdot {binom {2n}{n}}.}

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

(2nn)∼4nπn.{displaystyle {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 2π{displaystyle {sqrt {2pi }}} in front of the Stirling formula, by comparison.

Simple bounds that immediately follow from 4n=(1+1)2n=∑k=02n(2nk){displaystyle 4^{n}=(1+1)^{2n}=sum _{k=0}^{2n}{binom {2n}{k}}} are

4n2n+1≤(2nn)≤4n for all n≥1{displaystyle {frac {4^{n}}{2n+1}}leq {2n choose n}leq 4^{n}{text{ for all }}ngeq 1}

Some better bounds are^[2]

4n4n≤(2nn)≤4n3n+1 for all n≥1{displaystyle {frac {4^{n}}{sqrt {4n}}}leq {2n choose n}leq {frac {4^{n}}{sqrt {3n+1}}}{text{ for all }}ngeq 1}

and, if more accuracy is required,

(2nn)=4nπn(1−cnn) where 19<cn<18{displaystyle {2n choose n}={frac {4^{n}}{sqrt {pi n}}}left(1-{frac {c_{n}}{n}}right){text{ where }}{frac {1}{9}}<c_{n}<{frac {1}{8}}} for all n≥1.{displaystyle ngeq 1.}^{[citation needed]}

The only central binomial coefficient that is odd is 1. More specifically, the number of factors of 2 in (2nn){displaystyle {binom {2n}{n}}} is equal to the number of ones in the binary representation of n.^[3]

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

The central binomial coefficient (2nn){displaystyle {2n choose n}} equals the sum of the squares of the elements in row n of Pascal's triangle.^[1]

Related sequences

The closely related Catalan numbers C_n are given by:

Cn=1n+1(2nn)=(2nn)−(2nn+1) for all n≥0.{displaystyle C_{n}={frac {1}{n+1}}{2n choose n}={2n choose n}-{2n choose n+1}{text{ for all }}ngeq 0.}

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

References

• 
  Koshy, Thomas (2008), Catalan Numbers with Applications, Oxford University Press, ISBN 978-0-19533-454-8.

External links

• 
  "Central binomial coefficient". PlanetMath.
  


• 
  "Binomial coefficient". PlanetMath.
  


• 
  "Pascal's triangle". PlanetMath.
  


• 
  "Catalan numbers". PlanetMath.
  

This article incorporates material from Central binomial coefficient on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.