Seems like (but needs checking that)
$$ \sum \frac{1}{\ell!} s(\ell,k) z^\ell t^k = \exp(t \sinh(z)). $$
That could probably be used to find other formulas, recurrences, etc.

ADDED:
http://oeis.org/A136630  OEIS sequence A136630 is about these numbers.