I have found several authoritative sources with differing methods for calculating the control chart factor $c_4$. Are these methods truly equivalent, and if they are not, which method is the "real" method for calculating this constant?
$$c_4=\sqrt{\frac{2}{n-1}}\frac{\left ( \frac{n-2}{2} \right )!}{\left ( \frac{n-3}{2} \right )!}$$(Neubauer, D. V. (Ed.) ASTM MNL7, Manual on Presentation of Data and Control Chart Analysis ASTM International, 2010) $$c_4=\frac{\sqrt{\frac{2}{n-1}}\times \Gamma \left [ \frac{n}{2} \right ]}{\Gamma \left [ \frac{n-1}{2} \right ]}$$(M.A. Mahmoud et al. "Estimating the Standard Deviation in Quality-Control Applications" published in the Journal of Quality Technology in Vol. 42, No. 4, October 2010.) $$c_4=\sqrt{\frac{2}{n-1}}\frac{\left ( \frac{n}{2}-1 \right )!}{\left ( \frac{n-1}{2}-1 \right )!}$$(http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc32.htm)
