Inverse laplace transform pf infinite product

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Maszenka
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I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?
Screenshot_20210513-190516_Messenger.jpg
 

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$$g(t) =\dfrac{ e^{-nt}}{(n-1)!} (e^t-1)^{n-1}$$

I suspect it can be proven using induction. Have a go.
 

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