Double sequence convergence Let be $$S(m,k)$$
  number of partitions of a $k$ element set into $m$ nonempty parts investigating with generating functions I get this very interesting equation$$\sum_{k=0}^ {\infty}\sum_{m=0}^ {\infty}S(m,k)\frac{1}{k!}=e^{e-1} $$
Can someone tell me if I am right.

The inner sum actually has a finite number of non-zero members:
$$
\sum_{m=0}^ {\infty}S(m,k)=
\sum_{m=0}^ {k}S(m,k)=B_k
$$
where $B_k$ are the Bell numbers.
The exponential generating function for them is 
$$\sum_{k=0}^\infty \frac{B_k x^k}{k!}=e^{e^x-1}.$$ Putting $x=1$ gives the required equality.
