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A330057 Number of set-systems covering n vertices with no singletons or endpoints.

%I A330057 #9 Jan 16 2023 14:49:31
%S A330057 1,0,0,5,1703,66954642,144115175199102143,
%T A330057 1329227995784915808340204290157341181,
%U A330057 226156424291633194186662080095093568664788471116325389572604136316742486364
%N A330057 Number of set-systems covering n vertices with no singletons or endpoints.
%C A330057 A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1).
%H A330057 Andrew Howroyd, <a href="/A330057/b330057.txt">Table of n, a(n) for n = 0..11</a>
%H A330057 Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>
%F A330057 Binomial transform is A330056.
%e A330057 The a(3) = 5 set-systems:
%e A330057   {{1,2},{1,3},{2,3}}
%e A330057   {{1,2},{1,3},{1,2,3}}
%e A330057   {{1,2},{2,3},{1,2,3}}
%e A330057   {{1,3},{2,3},{1,2,3}}
%e A330057   {{1,2},{1,3},{2,3},{1,2,3}}
%t A330057 Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]
%o A330057 (PARI) \\ here b(n) is A330056(n).
%o A330057 AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
%o A330057 b(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-(n-k)-1) * sum(j=0, k\2, sum(i=0, k-2*j, binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) )))}
%o A330057 a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*b(n-k))} \\ _Andrew Howroyd_, Jan 16 2023
%Y A330057 The version for non-isomorphic set-systems is A330055 (by weight).
%Y A330057 The non-covering version is A330056.
%Y A330057 Set-systems with no singletons are A016031.
%Y A330057 Set-systems with no endpoints are A330059.
%Y A330057 Non-isomorphic set-systems with no singletons are A306005 (by weight).
%Y A330057 Non-isomorphic set-systems with no endpoints are A330054 (by weight).
%Y A330057 Non-isomorphic set-systems counted by vertices are A000612.
%Y A330057 Non-isomorphic set-systems counted by weight are A283877.
%Y A330057 Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.
%K A330057 nonn
%O A330057 0,4
%A A330057 _Gus Wiseman_, Nov 30 2019
%E A330057 Terms a(5) and beyond from _Andrew Howroyd_, Jan 16 2023