%I #20 Jan 01 2020 10:29:38
%S 1,2,10,82,1038,19754,561778,23890766,1516425978,142478603490,
%T 19560464078774,3868751287074546,1088233853378616578,
%U 430599111941369628326,237480490462200909980594,181131722604060126010422898,189780362331001773747253412782,271553393666987988551182068682458,527932854364810523962111033565618786
%N Total number of subsets of X that are both open and closed summed over all distinct topological spaces X that can be placed on an n-set.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Clopen_set">Clopen set</a>.
%F E.g.f.: A(exp(x) - 1)^2 where A(x) is the e.g.f. for A001035.
%F a(n) = Sum_{k=1..n} A247232(n,k)*2^k.
%e a(2) = 10. Let X = {a,b}. There are four distinct topologies (A000798) that can be placed on X: {{},X} {{},{a},X} {{}, {b},X} {{},{a},{b},X}. These topologies have 2 + 2 + 2 + 4 sets respectively that are both open and closed.
%t A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t lg = Length[A001035];
%t A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
%t CoefficientList[A[Exp[x] - 1]^2 + O[x]^lg, x]*Range[0, lg - 1]! (* _Jean-François Alcover_, Jan 01 2020 *)
%Y Cf. A000798, A001035, A247232.
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Jan 23 2017