While computing conditional expectations of certain functionals of a Poisson white noise field (details are long and probably irrelevant), I've stumbled upon the need to use the following identity involving Stirling numbers of the second kind:
$$
\ell{k\brace \ell} = \sum_{j=\ell}^k {k\choose j-1} (-1)^{k-j} {j\brace \ell}.
$$
I used Manuel Kauers' [*Stirling*][1] package in order to produce a recurrence relation from which the identity can be easily proved. I still wonder, however, whether this is actually well-known, or there is some short proof...


  [1]: http://www.kauers.de/software.html