The Bernoulli Numbers Come from a Shift Operator

The Bernoulli numbers were defined by Faulhaber in terms of the following generating series. \[\frac{x}{e^x-1} = \sum_{k \geq 0} B_k \frac{x^k}{m!}\] But why? Where did this come from? Well, the mathematicians of the time were contemplating the following sorts of patterns: \[\begin{aligned} 1+2+\cdots+n &= \frac{n(n+1)}{2} \\ 1^2+2^2+\cdots+n^2 & = \frac{n(n+1)(2n+1)}{6} \\ 1^3+2^3+\cdots+n^3 & =…