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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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It appears to be true for $x > .8305123339$ approximately: $c_x \to 0$ as $x \to 0+$.

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    I plotted $f(x) = x \left( 1-{{2\,x}\choose x}\sqrt {\pi \,x}{4}^{-x} \right)$, and solved $f(x) = 1/9$ numerically.2012-12-17