%I #8 Jan 21 2023 16:12:01

%S 0,1,6,65,3297,2537672,412184904221,4132070624893905681577,

%T 174224571863520492218909428465944685216436,

%U 133392486801388257127953774730008469745829658368044283629394202488602260177922751

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

%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 set-systems with minimum covered vertex-degree 1.

%H Andrew Howroyd, <a href="/A327228/b327228.txt">Table of n, a(n) for n = 0..12</a>

%F Binomial transform of A327229.

%F a(n) = A058891(n+1) - A330059(n). - _Andrew Howroyd_, Jan 21 2023

%e The a(2) = 6 set-systems:

%e {{1}}

%e {{2}}

%e {{1,2}}

%e {{1},{2}}

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

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

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

%Y The covering version is A327229.

%Y The specialization to simple graphs is A245797.

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

%Y Cf. A058891, A059167, A327098, A327103, A327104, A327107, A327197, A327227, A327230, A330059.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 01 2019

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