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\}\{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:

:$C\_n\; =\; frac\{1\}\{n+1\}\; \{2n\; choose\; n\}.$

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

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