# Central binomial coefficient

Central binomial coefficient

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

: $\left\{2n choose n\right\} = frac\left\{\left(2n\right)!\right\}\left\{\left(n!\right)^2\right\}.$

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, &hellip;

These numbers have the generating function

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

By Stirling's formula we have

: $\left\{2n choose n\right\} sim frac\left\{4^n\right\}\left\{sqrt\left\{pi n$ as $n ightarrowinfty$.

Some useful bounds are

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

and, if more accuracy is required,

:$frac\left\{4^n\right\}\left\{sqrt$22 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:

:$C_n = frac\left\{1\right\}\left\{n+1\right\} \left\{2n choose n\right\}.$

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

ee also

* Erdős Squarefree Conjecture

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