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For $n\in\mathbb{Z}_0,m\in\mathbb{Z}^+$,

$n!_m=n\underbrace{!\cdots!}_{\text{m}}=\displaystyle\prod_{k=0}^{\lceil{n/m-1}\rceil}(n-km)$

Using this formula, we can indeed find the m-th multi factorial of n.

However, I have no idea how to prove it.

I found this at the following

In layman's terms, what is a multifactorial? - $x\underbrace{!!\cdots!}_{n\text{ times} }$

Please help me.

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  • $\begingroup$ Can you prove it for $m=2$? $\endgroup$ Feb 25 at 3:10
  • $\begingroup$ Getting induction vibes $\endgroup$ Feb 25 at 3:32
  • $\begingroup$ Please tell me.When $m=2$, if $n$ is even $n!!=2^nn!$, else if $n$ is odd, $n!!=(2k)!/(2^kk!)$. This is easy. However I can't prove it. $\endgroup$
    – sakura
    Feb 25 at 5:52
  • $\begingroup$ this is the definition $\endgroup$
    – Jakobian
    Feb 25 at 20:09
  • $\begingroup$ If this is the definition, tell me the citiation. $\endgroup$
    – sakura
    Feb 25 at 22:48

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