Formula for sum of combinations I am trying to find a closed-form formula or, at least, a different and more useful representation for the following sum of combinations:
\begin{equation}
\sum_{i=1}^{n}\frac{n!}{i!(n-i)!}\times\frac{(-1)^i}{i}
\end{equation}

We have that
$$\sum_{i=1}^{n}\frac{n!}{i!(n-i)!}\times\frac{(-1)^i}{i}=\sum_{i=1}^{n}\frac{(-1)^i}{i}\binom{n}{i}$$
then refer to


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*Sum of Pascal's triangle column

*Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$
