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I have a discrete probability distribution where $P(X=x)=(Factorial[n]/(n^x))*StirlingS2[x-1,n-1]$. So, when trying to calculate its mean with the following property: $\mu = \sum_{i} x_i \cdot P(x_i)$, I insert this expression in Mathematica:

Sum[(Factorial[n]/(n^x))*StirlingS2[x - 1, n - 1]*x, {x, 0, Infinity}]

But the program doesn't output the correct answer, which in this case would be n*HarmonicNumber[n], instead it provides: $\sum _{x=0}^{\infty } x n! n^{-x} \mathcal{S}_{x-1}^{(n-1)}$ as output.

I have tried to encapsulate the operation inside FunctionExpand and FullSimplify but I can´t make it work.

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Using:

$$\mathcal{S}_{x-1}^{(n-1)}=\frac{\sum _{j=0}^{n-1} (-1)^{j+n-1} \binom{n-1}{j} j^{x-1}}{(n-1)!}$$

we have:

FullSimplify[Sum[Sum[(Factorial[n]/(n^x))*((1/(n - 1)!) *(-1)^(j + n - 1)  Binomial[n - 1, j]  j^(x - 1))*x // FullSimplify, {x, 0, Infinity}] // FunctionExpand, {j, 0, n - 1}], Assumptions -> {n \[Element] PositiveIntegers}]

(*n HarmonicNumber[n]*)
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