How do you solve $$\Gamma(2n)=\Gamma(n)^n$$ The trivial solution is of course $n=1$, but there is another one according to WolframAlpha and it gives the numerical approximation
$$n ≈ 0.323782241917058...$$
How would one solve for the two solutions? The equation is a bit arbitrary and there's probably not a closed-form for the second solution, but I just want to see...maybe there is?
I doubt that it helps, but I tried to take the natural logarithm on both sides so we get:
\begin{align} \ln(\Gamma(2n)) &= \ln(\Gamma(n)^n) \\\\ & = n\ln(\Gamma(n))\end{align}
...but I don't know how to take it from here. Maybe use some series representation for $\ln\Gamma(n)$?