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A330056 Number of set-systems with n vertices and no singletons or endpoints.

%I A330056 #12 Jan 16 2023 14:49:35
%S A330056 1,1,1,6,1724,66963208,144115175600855641,
%T A330056 1329227995784915809349010517957163445,
%U A330056 226156424291633194186662080095093568675422295082604716043360995547325655259
%N A330056 Number of set-systems with n vertices and no singletons or endpoints.
%C A330056 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 A330056 Andrew Howroyd, <a href="/A330056/b330056.txt">Table of n, a(n) for n = 0..11</a>
%H A330056 Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>
%F A330056 Binomial transform of A330057.
%F A330056 a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} Sum_{i=0..k-2*j} (-1)^k * binomial(n,k) * 2^(2^(n-k)-(n-k)-1) * binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) where AS2(n,k) are the associated Stirling numbers of the 2nd kind (A008299). - _Andrew Howroyd_, Jan 16 2023
%e A330056 The a(3) = 6 set-systems:
%e A330056   {}
%e A330056   {{1,2},{1,3},{2,3}}
%e A330056   {{1,2},{1,3},{1,2,3}}
%e A330056   {{1,2},{2,3},{1,2,3}}
%e A330056   {{1,3},{2,3},{1,2,3}}
%e A330056   {{1,2},{1,3},{2,3},{1,2,3}}
%t A330056 Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]
%o A330056 (PARI) \\ Here AS2(n,k) is A008299 (associated Stirling of 2nd kind)
%o A330056 AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
%o A330056 a(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)) )))} \\ _Andrew Howroyd_, Jan 16 2023
%Y A330056 The version for non-isomorphic set-systems is A330055 (by weight).
%Y A330056 The covering case is A330057.
%Y A330056 Set-systems with no singletons are A016031.
%Y A330056 Set-systems with no endpoints are A330059.
%Y A330056 Non-isomorphic set-systems with no singletons are A306005 (by weight).
%Y A330056 Non-isomorphic set-systems with no endpoints are A330054, (by weight).
%Y A330056 Non-isomorphic set-systems counted by vertices are A000612.
%Y A330056 Non-isomorphic set-systems counted by weight are A283877.
%Y A330056 Cf. A007716, A055621, A008299, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.
%K A330056 nonn
%O A330056 0,4
%A A330056 _Gus Wiseman_, Nov 30 2019
%E A330056 Terms a(5) and beyond from _Andrew Howroyd_, Jan 16 2023