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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

pjmathematician
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  • Look at my answer: https://math.stackexchange.com/questions/3689757/proving-pi-27s-36-8-sqrt3-where-s-sum-n-0-infty-frac-left-left/3689772#3689772 – Jan Eerland May 26 '20 at 17:44
  • @Jan Thanks, but I want the applications, i had evaluated it by using a similar approach as yours – pjmathematician May 26 '20 at 17:47
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    $\wr$ gives $\wr$ – saulspatz May 26 '20 at 17:53
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    http://oeis.org/A056040 lists some applications – saulspatz May 26 '20 at 17:54
  • @saulspatz thanks! – pjmathematician May 26 '20 at 18:26
  • This is interesting. As someone who doesn't know anything about this, two questions: it is possible to analytically extend $\wr$ to the whole line just as $\Gamma$ extends $!$? If so, does ${\rm e}\wr$ actually coincide with swinging constant, i.e., is ${\rm e}\wr$ more than just a notation? – Ivo Terek May 26 '20 at 18:36
  • @Ivo Terek the first question is a nice question, by looking at the uses/application/equivalent definitions we see that $n\wr$ is mostly used as a series coefficient or used in number theory as it yields only natural numbers. For the second question, I don't think just substituting value of e will do the work, the analytical extension will help us do that. – pjmathematician May 27 '20 at 00:01

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