Swinging factorial and swinging constant

Question

The Swinging factorial $n≀$ defined as $$n≀=\frac{n!}{\left\lfloor{n/2}\right\rfloor!^2}$$ is relatively common and I found some results on Google. But when $$\sum_{n=0}^{\infty}\frac{1}{n≀}$$is calculated(by converting it into gamma and applying beta function) we get $\frac{8\pi\sqrt3}{27} + \frac{4}{3}$ which is quite a peculiar result and defining a constant(swinging constant)-$$e≀=\frac{8\pi\sqrt3}{27} + \frac{4}{3}$$ I wanted to ask that if there are any significant applications for both the swinging factorial and constant