A247232 -id:A247232 - cp's OEIS frontend

A281547 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.

1, 2, 10, 82, 1038, 19754, 561778, 23890766, 1516425978, 142478603490, 19560464078774, 3868751287074546, 1088233853378616578, 430599111941369628326, 237480490462200909980594, 181131722604060126010422898, 189780362331001773747253412782, 271553393666987988551182068682458, 527932854364810523962111033565618786

Offset: 0

Geoffrey Critzer, Jan 23 2017

nonn

			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.

Wikipedia, Clopen set.

Cf. A000798, A001035, A247232.

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
CoefficientList[A[Exp[x] - 1]^2 + O[x]^lg, x]*Range[0, lg - 1]! (* Jean-François Alcover, Jan 01 2020 *)

E.g.f.: A(exp(x) - 1)^2 where A(x) is the e.g.f. for A001035.

a(n) = Sum_{k=1..n} A247232(n,k)*2^k.

cp's OEIS Frontend

A281547 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.

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