Formula for connected graphs with n vertices

Question

In graph theory, is there a formula for the following:

How many simple graphs with n vertices exist such that the graph is connected?

Answer 1

There is a recursive way to find it, this idea is treated in the following book.

Let us denote the number in question by $f(n)$. Then $2^{\binom{n}{2}}=\sum_{k=1}^{n}\binom{n-1}{k-1}f(k)\cdot2^{\binom{n-k}{2}}$. this idea comes from selecting a special vertex and classifying all the graphs on aset of $n$ vertices depending on the size of the component containing that special vertex. Using this we compute a few cases:

$f(1)=1,f(2)=1,f(3)=4,f(4)=28,f(5)=728$ and $f(6)=26704$

I plugged these numbers into oeis and it gave me this sequence, however that sequence doesn't give any other formulas, it seems to give the same one I gave you, and an exponential generating function, but nothing juicy :)