This is a problem from Concrete Mathematics (Eq. 9.57 and 9.58 in 1995 edition).
$G(z) = \sum_{k}g_k z^k = e^{\sum_{k} \frac{z^k}{k^2}}$ is a generating function of sequence $g_k$. By taking the derivative wrt $z$ we get G'(z) = \sum_{k} \frac{z^{k-1}}{k} G(z) and at the same time of course G'(z) = \sum_k k g_k z^{k-1}. Coefficient at $z^{n-1}$ term would be obviously $n g_n$. What I can't sort out why coefficient on the other side of the equation is $\sum_{k=1}^{n-1} \frac{g_k}{n-k}$. I were trying to derive some convolution by expanding the exponential function, but couldn't quite make it.
Please don't solve it, some hints will do.