How many connected graphs with $n$ nodes are there approximately?

Question

OEIS states the the number of graphs with $0,1,2,...,19$ nodes is $$1, 1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476,$$ $$50335907869219, 29003487462848061, 31397381142761241960, 63969560113225176176277,$$ $$245871831682084026519528568, 1787331725248899088890200576580,$$ $$24636021429399867655322650759681644$$

• Is there a good approximation formula, in particular for large numbers ?

Answer 1

The probability of a random graph being disconnected or having a nontrivial automorphism is very small for large $n$, so a good approximation is just $$\frac{2^{n(n−1)/2}}{n!}...$$that is, the number of graphs on $n$ labeled nodes, divided by the maximum number of inequivalent permutations of the labels. Note that there are two approximations being made, which partially cancel each other out: taking symmetries into account increases this number, and removing disconnected graphs decreases it.

For $n=19$, this differs from the true number of connected graphs by about $0.1\%$.

Answer 2

Since you looked at A001349, presumably you saw this in the Formula section:

For asymptotics see Lupanov 1959, 1960, also Turner and Kautz, p. 18. - N. J. A. Sloane, Apr 08 2014

Have you followed up those references?