Operationally, using the generalized Dobinski formula in the [MO-Q][1] on the o.g.f. for the Bell polynomials $\phi_m(x) = \sum_{k=0}^m S(m,k) x^k$,

$$ \phi_{n+m}(x) = (\phi.(x))^{n+m} = e^{-x} \sum_{j =0}^{\infty} j^{n+m} \frac{x^j}{j!} = \sum_{j=0}^{\infty} (-1)^j [\sum_{k=0}^j (-1)^k \binom{j}{k} k^{n+m}] \frac{x^j}{j!} \; . $$


  [1]: http://mathoverflow.net/questions/172955/ordinary-generating-function-for-bell-numbers/172969#172969