Central binomial coefficient

Central binomial coefficient

In mathematics the "n"th central binomial coefficient is defined in terms of the binomial coefficient by

: {2n choose n} = frac{(2n)!}{(n!)^2}.

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 OEIS|id=A000984:

:num|1, num|2, num|6, num|20, num|70, 252, 924, 3432, 12870, 48620, …

These numbers have the generating function

:frac{1}{sqrt{1-4x = 1 + 2x + 6x^2 + 20x^3 + 70x^4 + 252x^5 + cdots.

By Stirling's formula we have

: {2n choose n} sim frac{4^n}{sqrt{pi n as n ightarrowinfty.

Some useful bounds are

:frac{4^n}{sqrt{4n leq {2n choose n} leq frac{4^n}{sqrt{3n+1 for all n geq 1

and, if more accuracy is required,

:frac{4^n}{sqrt22 over 7}n + {6 over 7} leq {2n choose n} leq frac{4^n}{sqrt28 over 9}n+{8 over 9} for all n geq 1.

The closely related Catalan numbers "C""n" are given by:

:C_n = frac{1}{n+1} {2n choose n}.

A slight generalization of central binomial coefficients is to take them as { m choose {lfloor frac{m}{2} floor} }and so the former definition is a particular case when "m" = 2"n", that is, when "m" is even.

ee also

* Erdős Squarefree Conjecture

External links


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