I doubt that more than https://oeis.org/A105599 is known.

The formula given in Bollobas is
$$
\frac{1}{m!}\sum_{j=0}^m \left(-\frac{1}{2}\right)^j \binom{m}{j}\binom{n-1}{m+j-1}n^{n-m-j}(m+j)!,
$$
where $n$ is the number of vertices and $m$ is the number of components.