%I #8 Jan 21 2023 16:11:57

%S 0,1,4,50,3069,2521782,412169726428,4132070622008664529903,

%T 174224571863520492185852863478334475199686,

%U 133392486801388257127953774730008469744261637221272599199572772174870315402893538

%N Number of set-systems covering n vertices with at least one endpoint/leaf.

%C Covering means there are no isolated vertices.

%C A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

%C Also covering set-systems with minimum vertex-degree 1.

%H Andrew Howroyd, <a href="/A327229/b327229.txt">Table of n, a(n) for n = 0..12</a>

%F Inverse binomial transform of A327228.

%e The a(2) = 4 set-systems:

%e {{1,2}}

%e {{1},{2}}

%e {{1},{1,2}}

%e {{2},{1,2}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,3}]

%Y The non-covering version is A327228.

%Y The specialization to simple graphs is A327227.

%Y The unlabeled version is A327230.

%Y BII-numbers of these set-systems are A327105.

%Y Cf. A003465, A245797, A327079, A327098, A327103, A327107, A327197.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 01 2019

%E Terms a(5) and beyond from _Andrew Howroyd_, Jan 21 2023