[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the Bell numbers [math]B_k=|P(k)|[/math], which are the moments of the Poisson law [math]p_1[/math], have the following properties:

[[math]] B_{k+1}=\sum_{r=0}^k\binom{k}{r}B_r\quad,\quad B_k=\frac{1}{e}\sum_{r=0}^\infty\frac{r^k}{r!} [[/math]]

[[math]] \sum_{k=0}^\infty\frac{B_k}{k!}\,z^k=e^{e^z-1}\quad,\quad B_k=\frac{k!}{2\pi ie}\int_{|z|=1}\frac{e^{e^z}}{z^{k+1}}\,dz [[/math]]

Also, prove as well that we have [math]\log B_k/k\simeq\log k-\log\log k-1[/math].