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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 $1/9<c_n<1/8$.

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

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  • $\begingroup$ Thanks. I'm mostly interested in $x>1$. Would you know of any reference relating to your statement? $\endgroup$ – Eckhard Dec 17 '12 at 20:09
  • $\begingroup$ I plotted $f(x) = x \left( 1-{{2\,x}\choose x}\sqrt {\pi \,x}{4}^{-x} \right)$, and solved $f(x) = 1/9$ numerically. $\endgroup$ – Robert Israel Dec 17 '12 at 20:51

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