MHB Inverse laplace transform pf infinite product

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The discussion revolves around finding the inverse Laplace transform of the infinite product represented by the function g(t) = (e^(-nt)/((n-1)!)) * (e^t - 1)^(n-1). Participants suggest that the solution may be approached through mathematical induction. There is a request for assistance in proving the inverse transform. The conversation emphasizes the need for a clear method to tackle the problem. Overall, the focus is on deriving the inverse Laplace transform effectively.
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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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