# Closed form for ∏∞n=12n√Γ(2n+12)Γ(2n)\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}

Is there a closed form for the following infinite product?

The beautiful idea of Raymond Manzoni can actually be made rigorous. Consider a finite product $\prod_{n=1}^{L}$ and take its logarithm. After using duplication formula for the gamma function and telescoping, it simplifies to the following:
This is an exact relation, valid for any $L$. Now it suffices to use Stirling,
So the answer is indeed $\displaystyle\frac{8\sqrt{\pi}}{e^2}$.