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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

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  • $\begingroup$ Look at my answer: math.stackexchange.com/questions/3689757/… $\endgroup$ Commented May 26, 2020 at 17:44
  • $\begingroup$ @Jan Thanks, but I want the applications, i had evaluated it by using a similar approach as yours $\endgroup$ Commented May 26, 2020 at 17:47
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    $\begingroup$ $\wr$ gives $\wr$ $\endgroup$
    – saulspatz
    Commented May 26, 2020 at 17:53
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    $\begingroup$ oeis.org/A056040 lists some applications $\endgroup$
    – saulspatz
    Commented May 26, 2020 at 17:54
  • $\begingroup$ @saulspatz thanks! $\endgroup$ Commented May 26, 2020 at 18:26

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