Central binomial coefficient

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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!)^2text 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)

Contents

• 
  1 Properties
  


• 
  2 Related sequences
  


• 
  3 References
  


• 
  4 External links
  

Properties

The central binomial coefficients have ordinary generating function

11−4x=1+2x+6x2+20x3+70x4+252x5+⋯displaystyle frac 1sqrt 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 2nnfrac x^nn!=e^2xI_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+2n+1cdot binom 2nn.

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

(2nn)∼4nπn.displaystyle 2n choose nsim frac 4^nsqrt 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 2πdisplaystyle 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=0^2nbinom 2nk are

4n2n+1≤(2nn)≤4n for all n≥1displaystyle frac 4^n2n+1leq 2n choose nleq 4^ntext for all ngeq 1

Some better bounds are^[2]

4n4n≤(2nn)≤4n3n+1 for all n≥1displaystyle frac 4^nsqrt 4nleq 2n choose nleq frac 4^nsqrt 3n+1text for all ngeq 1

and, if more accuracy is required,

(2nn)=4nπn(1−cnn) where 19<cn<18displaystyle 2n choose n=frac 4^nsqrt pi nleft(1-frac c_nnright)text where frac 19<c_n<frac 18 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 2nn 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 1n+12n choose n=2n choose n-2n choose n+1text 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 1nmathrm 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

1. 
   ^ ^ab Sloane, N.J.A. (ed.). "Sequence A000984 (Central binomial coefficients)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
   


2. 
   ^ Kazarinoff, N.D. Geometric inequalities, New York: Random House, 1961
   


3. 
   ^ Sloane, N.J.A. (ed.). "Sequence A000120". 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 .

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.

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