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Central binomial coefficient

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

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

PropertiesEdit

The central binomial coefficients have ordinary generating function

 

and exponential generating function

 

where I0 is a modified Bessel function of the first kind.[1]

They also satisfy the recurrence

 

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

 

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   in front of the Stirling formula, by comparison.

Simple bounds that immediately follow from   are

 

Some better bounds are[2]

 

and, if more accuracy is required,

  for all  [citation needed]

The only central binomial coefficient that is odd is 1. More specifically, the number of factors of 2 in   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   equals the sum of the squares of the elements in row n of Pascal's triangle.[1]

Related sequencesEdit

The closely related Catalan numbers Cn are given by:

 

A slight generalization of central binomial coefficients is to take them as  , with appropriate real numbers n, where   is the gamma function and   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.

ReferencesEdit

  1. ^ a b 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 linksEdit