<s>[INST] Compute a sum with partial fraction decomposition and generating functions I am trying to compute $$\sum_{k=1}^{n-1} \frac{1}{k(n-k)}$$ using a term by term partial fraction decomposition and also with generating functions, but I'm stuck. 
 [/INST] Note that:
$$
\frac{1}{k (n - k)} = \frac{1}{k n} + \frac{1}{n (n - k)}
$$
Thus your sum is:
$\begin{align}
\sum_{1 \le k \le n - 1} \frac{1}{k (n - k)}
  &= \sum_{1 \le k \le n - 1} \frac{1}{k n}
       + \sum_{1 \le k \le n - 1} \frac{1}{n (n - k)} \\
  &= \frac{1}{n} \sum_{1 \le k \le n - 1} \frac{1}{k}
       + \frac{1}{n} \sum_{1 \le k \le n - 1} \frac{1}{n - k} \\
  &= \frac{2}{n} \sum_{1 \le k \le n - 1} \frac{1}{k} \\
  &= \frac{2}{n} H_{n - 1}
\end{align}$
</s>