Let $X$ be the random variable uniformly distributed on $\{0,\dots,d-1\}$. This is, $$\operatorname{Prob}(X = i) = \frac{1}{d}$$ for $i \in \{0,\dots,d-1\}$. Computations suggest that the cumulants of $X$ are, for $n \geq 2$, given by $$\kappa_n(X) = \frac{B_n}{n} (d^n-1)$$ where $B_n$ is the $n$-th Bernoulli number.
Where can I find this stated in the presented form?
The moments are given by $$m_n(X) = \frac{1}{d}\sum_{k=0}^{d-1} k^n = \frac{1}{d}\sum_{j=0}^n(-1)^j B_j \binom{n}{j}\frac{(d-1)^{n-j+1}}{n-j+1}.$$
How can the above formula be derived from this?