%I #15 Jan 16 2023 14:49:20

%S 1,1,2,63,29471,2144945976,9223371624669871587,

%T 170141183460469227599616678821978424151,

%U 57896044618658097711785492504343953752410420469299789800819363538011879603532

%N Number of set-systems with n vertices and no endpoints.

%C A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1).

%H Andrew Howroyd, <a href="/A330059/b330059.txt">Table of n, a(n) for n = 0..11</a>

%F a(n) = Sum_{k=0..n} Sum(j=0..k} (-1)^k * binomial(n,k) * 2^(2^(n-k)-1) * Stirling2(k,j) * 2^(j*(n-k)). - _Andrew Howroyd_, Jan 16 2023

%e The a(2) = 2 set-systems are {} and {{1},{2},{1,2}}. The a(3) = 63 set-systems are:

%e 0 {2}{3}{12}{13} {1}{3}{12}{13}{23}

%e {1}{2}{12} {2}{12}{13}{23} {2}{3}{12}{13}{23}

%e {1}{3}{13} {2}{3}{12}{123} {1}{2}{12}{23}{123}

%e {2}{3}{23} {2}{3}{13}{123} {1}{2}{13}{23}{123}

%e {12}{13}{23} {3}{12}{13}{23} {1}{3}{12}{13}{123}

%e {1}{23}{123} {1}{13}{23}{123} {1}{3}{12}{23}{123}

%e {2}{13}{123} {2}{12}{13}{123} {1}{3}{13}{23}{123}

%e {3}{12}{123} {2}{12}{23}{123} {2}{3}{12}{13}{123}

%e {12}{13}{123} {2}{13}{23}{123} {2}{3}{12}{23}{123}

%e {12}{23}{123} {3}{12}{13}{123} {2}{3}{13}{23}{123}

%e {13}{23}{123} {3}{12}{23}{123} {1}{12}{13}{23}{123}

%e {1}{2}{13}{23} {3}{13}{23}{123} {2}{12}{13}{23}{123}

%e {1}{2}{3}{123} {12}{13}{23}{123} {3}{12}{13}{23}{123}

%e {1}{3}{12}{23} {1}{2}{3}{12}{13} {1}{2}{3}{12}{13}{23}

%e {1}{12}{13}{23} {1}{2}{3}{12}{23} {1}{2}{3}{12}{13}{123}

%e {1}{2}{13}{123} {1}{2}{3}{13}{23} {1}{2}{3}{12}{23}{123}

%e {1}{2}{23}{123} {1}{2}{12}{13}{23} {1}{2}{3}{13}{23}{123}

%e {1}{3}{12}{123} {1}{2}{3}{12}{123} {1}{2}{12}{13}{23}{123}

%e {1}{3}{23}{123} {1}{2}{3}{13}{123} {1}{3}{12}{13}{23}{123}

%e {1}{12}{13}{123} {1}{2}{3}{23}{123} {2}{3}{12}{13}{23}{123}

%e {1}{12}{23}{123} {1}{2}{12}{13}{123} {1}{2}{3}{12}{13}{23}{123}

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]

%o (PARI) a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-1)*sum(j=0, k, stirling(k,j,2)*2^(j*(n-k)) ))} \\ _Andrew Howroyd_, Jan 16 2023

%Y The case with no singletons is A330056.

%Y The unlabeled version is A330054 (by weight) or A330124 (by vertices).

%Y Set-systems with no singletons are A016031.

%Y Non-isomorphic set-systems with no singletons are A306005 (by weight).

%Y Cf. A000612, A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330057, A330058.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 01 2019

%E Terms a(5) and beyond from _Andrew Howroyd_, Jan 16 2023