How many sublattices does the powerset lattice $2^n$ contain for $n$ finite? (up to equality, not isomorphism)


I thought for sure this would be easy to find on OEIS, but so far I am coming up empty.


I really am interested in seeing a list of small examples, say up to $n=5$ or $6$ maybe, although perhaps already things blow up too much at that point for this to be feasible. Ideally there would be a systematic way to write down examples, but just as an entryway to the literature, I thought I'd ask this as a counting question.



How many sublattices are contained in the powerset lattice of a finite set?
[OEIS A306445:](https://oeis.org/A306445) 2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, 92294021880898055590522262, 96307116899378725213365550192, 137362837456925278519331211455157, 266379254536998812281897840071155592


Number of collections of subsets of $\{1, 2,\ldots,n\}$ that are closed under union and intersection (starting at $n=0$).



This count includes the empty sublattice, so if you don't want that you should subtract one from every term.




