<s>[INST] Approximating factorial using identity $\frac{1}{x}!\frac{2}{x}!\cdots\cdot\frac{x}{x}!=\frac{ {x}!\cdot(2\pi)^{\frac{x-1}{2}} }{ x^x\cdot\sqrt{x} }$ I created a function that describes the product of the inverse multiples of a factorial
$$ m(x) = \frac{1}{x}!\cdot\frac{2}{x}!\cdot\frac{3}{x}!\cdots\frac{x-1}{x}!\cdot\frac{x}{x}!$$
for some reasons i thought this function might be useful, thats why i'm posting an incomplete explanation to the best of my knowledge but after some rigorous calculation, i was able to express $m(x)$ as a formula
$$ m(x) = \frac{ {x}!\cdot(2\pi)^{\frac{x-1}{2}} }{ x^x\cdot\sqrt{x} }$$
it turns out that the graph of $m(x)$ is a very simple one, makes me suggest it would be easy to interpolate and create a super factorial approximation from it
$$ {x}! = x^x\cdot\sqrt{x}\cdot\sqrt{2\pi}\cdot m(x)\cdot\sqrt{{2\pi}^{-x}}$$
and it resembles striling's approximation
$${x}! \approx x^x\cdot\sqrt{x}\cdot\sqrt{2\pi}\cdot e^{-x} $$

$$
m(1)=1, m(2)=\frac{1}{2}!, m(3)=\frac{4\pi}{3^{5/2}}, m(4)=\frac{3\pi^{3/2}}{2^{9/2}}, m(5)=\frac{96\pi^2}{5^{9/2}}, m(6)=\frac{45\pi^{5/2}}{3^{13/2}} , m(7)=\cdots
$$
now my question, can you help with the approximation, does the function gives any more information
 [/INST] Here is the proof of how I got $m(x)$
and since we're till speaking on approximation i also want to share this to you (degrees) @claude leibovici
$$ \sin{5} \approx \frac{1}{6+\sqrt{3}+\sqrt{14}}$$
The factorial function above is actually extended to the gamma function, but i am using factorial notation for simplicity $ \Gamma(x) = (x-1)!$ and i was able to write out the $m(x)$ function using help from euler reflextion formula  check here
$$ {x}!(-x)! = \frac{\pi x }{\sin{\pi x }}$$
I reduced each term in the factorial of $m(x)$ so that i would always get it's negative face by using ${x}!=x(x-1)!$ and ignore product of unity there
$$
\begin{array}
\\
m(1) = 1\\
m(2) = \frac{1}{2}!\\
m(3) = \frac{1}{3}!\cdot\frac{2}{3}!\\
m(3) = \frac{1}{3}!\cdot\frac{-1}{3}!\cdot\frac{2}{3}\\
m(4) = \frac{1}{4}!\cdot\frac{2}{4}!\cdot\frac{3}{4}!\\
m(4) = \frac{1}{4}!\cdot\frac{-1}{4}!\cdot\frac{2}{4}!\cdot\frac{3}{4}\\
m(5) = \frac{1}{5}!\cdot\frac{2}{5}!\cdot\frac{3}{5}!\cdot\frac{4}{5}!\\
m(5) = \frac{1}{5}!\cdot\frac{-1}{5}!\cdot\frac{2}{5}!\cdot\frac{-2}{5}!\cdot\frac{3}{5}\cdot\frac{4}{5}\\
m(6) = \frac{1}{6}!\cdot\frac{2}{6}!\cdot\frac{3}{6}!\cdot\frac{4}{6}!\cdot\frac{5}{6}!\\
m(6) = \frac{1}{6}!\cdot\frac{-1}{6}!\cdot\frac{2}{6}!\cdot\frac{-2}{6}!\cdot\frac{3}{6}!\cdot\frac{4}{6}\cdot\frac{5}{6}\\
m(x) = \frac{1}{x}!\cdot\frac{2}{x}!\cdot\frac{3}{x}!\cdot\frac{4}{x}\cdots\frac{x-1}{x}!\\
m(x) = \frac{1}{x}!\cdot\frac{-1}{x}!\cdot\frac{2}{x}!\cdot\frac{-2}{x}!\cdot\frac{3}{x}!\cdot\frac{-3}{x}!\cdots\frac{n}{x}!\cdot\frac{-n}{x}!\cdots\cdots\frac{x-n}{x}\cdots\frac{x-3}{x}\cdot\frac{x-2}{x}\cdot\frac{x-1}{x}\cdot f(n)\\
\end{array}
$$
above, $n$ represents a function of the variable $x$ and $f(n)$ is a function of $n$
$$n= 
\begin{cases}
\frac{x}{2}-1, &\text{if $x$ is even}\\
\frac{x+1}{2}-1, &\text{if $x$ is odd}
\end{cases}
$$
$$f(n)=
\begin{cases}
1, &\text{if $x$ is odd}\\
\frac{1}{2}!, &\text{if $x$ is even}
\end{cases}
$$
$\frac{1}{2}! = m(2)$ and from the $m(x)$ formula it is directly equal to $\frac{\sqrt{\pi}}{2}$
$m(x)$ is variant, so i'll solve each part of it separately and apply some trigonometry identity to simplify
$$\frac{1}{x}!\cdot\frac{-1}{x}!\cdot\frac{2}{x}!\cdot\frac{-2}{x}!\cdot\frac{3}{x}!\cdot\frac{-3}{x}!\cdots\frac{n}{x}!\cdot\frac{-n}{x}! = 
\frac{\frac{\pi}{x} }{\sin{\frac{\pi}{x} }}\cdot\frac{ \frac{2\pi}{x} }{\sin{\frac{2\pi}{x} }}\cdot\frac{\frac{3\pi}{x} }{\sin{\frac{3\pi}{x} }}\cdots\frac{\frac{n\pi}{x} }{\sin{\frac{n\pi}{x} }}$$
$$\frac{1}{x}!\cdot\frac{-1}{x}!\cdot\frac{2}{x}!\cdot\frac{-2}{x}!\cdot\frac{3}{x}!\cdot\frac{-3}{x}!\cdots\frac{n}{x}!\cdot\frac{-n}{x}! = \frac{ {n}!\cdot \pi^{n}}{ x^n \cdot (\sin{\frac{\pi}{x}}\cdot\sin{\frac{2\pi}{x}}\cdot\sin{\frac{3\pi}{x}}\cdots\sin{\frac{n\pi}{x}}) }$$
$$\frac{1}{x}!\cdot\frac{-1}{x}!\cdot\frac{2}{x}!\cdot\frac{-2}{x}!\cdot\frac{3}{x}!\cdot\frac{-3}{x}!\cdots\frac{n}{x}!\cdot\frac{-n}{x}! = \frac{{n}!\cdot \pi^n\cdot \sqrt{2^{x-1}}}{x^n\cdot \sqrt{x}}$$
$$\frac{x-n}{x}\cdots\frac{x-3}{x}\cdot\frac{x-2}{x}\cdot\frac{x-1}{x}\cdot f(n) = { (x-n)\cdots(x-3)(x-2)(x-1) }\cdot \frac{f(n)}{x^n}$$
$${x}! = x(x-1)(x-2)(x-3)\cdots(x-n)\cdot(x-n-1)!$$
$$(x-1)(x-2)(x-3)\cdots(x-n) = \frac{{x}!}{x(x-n-1)!}$$
$$\frac{x-n}{x}\cdots\frac{x-3}{x}\cdot\frac{x-2}{x}\cdot\frac{x-1}{x}\cdot f(n) = \frac{ {x}!f(n)}{x(x-n-1)!x^n}$$
$$m(x) =  \frac{{n}!\cdot\pi^n\cdot\sqrt{2^{x-1}}\cdot{x}!\cdot f(n)}{x^{2n}\cdot\sqrt{x}\cdot x\cdot (x-n-1)!}$$
if we simplify further to remove the $n$ and $f(n)$, we would arrive at the formula i wrote in the question
</s>