number of different undirected graphs?
Undirected graphs with self-loops on $n$ vertices have $\binom{n}{2} + n = \binom{n+1}{2}$ potential edges. Therefore the are $2^{\binom{n+1}{2}}$ different graphs. For $n = 1,2,3,\ldots$, this is
$$ 2^1, 2^3, 2^6, 2^{10}, 2^{15}, 2^{21}, 2^{28}, 2^{36}, 2^{45}, 2^{55}, \ldots $$