Table 1

$n!~\frac{\sqrt{2\pi }{e}^{-n}{n}^{n+1}}{\sqrt{n-1/6}}={\delta }_n$

(Batir [24])

$n!~\sqrt{\left(2n+1/3\right)\pi }{\left(n/e\right)}^n={\gamma }_n$

(Gosper [25])

$n!~\sqrt{\pi }\sqrt[6]{8n^3+4n^2+n+1/30}{\left(n/e\right)}^n={\rho }_n$

(Ramanujan [26])