Does $\sum_{n=1}^\infty \frac{T(n)}{2^{2^n}}$ converge?

Question

Let $T(n)$ be the number of distinct topologies on a set with $n$ elements. Does $\displaystyle\sum_{n=1}^\infty \displaystyle\frac{T(n)}{2^{2^n}}$ converge?

There is not much context to this unfortunately. It's a problem I came up with myself, when counting the number of topologies on an $n$-element set for $n=2,3$ (I am a beginner in topology). I am not sure of the difficulty of this problem but any progress toward a solution would be appreciated.

Answer 1

I believe the answer is yes. From this paper: The Number of Finite Topologies- D. Kleitman and B. Rothschild, the authors show that $T(n)$ is like $O(2^{n^2/4})$.