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On Wikipedia I came across the following equation for the central binomial coefficients: $$ \binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right) $$ for some $1/9.

Does anyone know of a better reference for this fact than wikipedia or planet math? Also, does the equality continue to hold for positive real numbers $x$ instead of the integer $n$ if we replace the factorials involved in the definition of the binomial coefficient by Gamma functions?

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    The answers in [this question](http://math.stackexchange.com/questions/58560/elementary-central-binomial-coefficient-estimates) might be helpful.2012-12-17

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