Asymptotic number of unlabeled graphs

If you apply Stirling's formula,

$c(n)\approx \frac{2^{n^2}e^{-n}}{n^n}$.

$\log c(n)\approx n^2 \log 2 - n \log n + n$

As the log increases faster than $n$, the growth is faster than any exponential.

See the OEIS, sequence A000088 for an asymptotic formula for the error. In particular, if $G(n)$ is the number of unlabeled graphs on n nodes, then

$$ {G(n) \over c(n)} = 1 + {n^2 - n \over 2^{n-1}} + O(n^2 2^{-2n}) $$

as $n \to \infty$. I suspect there is a simple combinatorial explanation for that first term but I don't know what it is; the source is Harary and Palmer, Graphical Enumeration, which may explain that.