You can generate the Bernoullis from graphical interpretations of surjection mappings presented in OEIS-[A133314][1]  (cf. [A049019][2], [A019538][3], [A008292][4]) weighted by the reciprocal integers. (See also MOQ-[61252][5].). 


  [1]: https://oeis.org/A133314
  [2]: https://oeis.org/A049019
  [3]: https://oeis.org/A019538
  [4]: https://oeis.org/A008292
  [5]: http://mathoverflow.net/questions/61252/why-do-bernoulli-numbers-arise-everywhere