What is the number of simple connected graphs that are sparse (few edges compared to the number of nodes)?

Question

The graphs that I consider are:

1. labelled (so, I do not want to count them up to isomorphism).
2. simple (no loops and at most one edge between two nodes).
3. connected.

In

How to calculate the number of possible connected simple graphs with $n$ labelled vertices

there is an answer to the count of such graphs given $n$ vertices. Would there be a formula of the number of such graphs if the number of vertices $n$ and the number of edges $m$ are given? My situation is that I need to estimate the number of such graphs when $n$ is large (say 3000), but $m$ is small.