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A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

%I A367916 #11 Dec 29 2023 16:41:17
%S A367916 1,2,6,45,1376,161587,64552473,85987037645,386933032425826,
%T A367916 6005080379837219319,328011924848834642962619,
%U A367916 64153024576968812343635391868,45547297603829979923254392040011994,118654043008142499115765307533395739785599
%N A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.
%H A367916 Andrew Howroyd, <a href="/A367916/b367916.txt">Table of n, a(n) for n = 0..50</a>
%F A367916 Binomial transform of A054780.
%e A367916 The a(0) = 1 through a(2) = 6 set-systems:
%e A367916   {}  {}     {}
%e A367916       {{1}}  {{1}}
%e A367916              {{2}}
%e A367916              {{1},{2}}
%e A367916              {{1},{1,2}}
%e A367916              {{2},{1,2}}
%t A367916 Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Length[Union@@#]==Length[#]&]],{n,0,3}]
%o A367916 (PARI) \\ Here b(n) is A054780(n).
%o A367916 b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(2^k-1, n))
%o A367916 a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ _Andrew Howroyd_, Dec 29 2023
%Y A367916 The covering case is A054780.
%Y A367916 For graphs we have A367862, covering A367863, unlabeled A006649.
%Y A367916 These set-systems have ranks A367917.
%Y A367916 A000372 counts antichains, covering A006126, nonempty A014466.
%Y A367916 A003465 counts set-systems covering {1..n}, unlabeled A055621.
%Y A367916 A058891 counts set-systems, unlabeled A000612.
%Y A367916 A059201 counts covering T_0 set-systems.
%Y A367916 A136556 counts set-systems on {1..n} with n edges.
%Y A367916 Cf. A092918, A102896, A133686, A306445, A323818, A355740, A367770, A367869, A367901, A367902, A367905.
%K A367916 nonn
%O A367916 0,2
%A A367916 _Gus Wiseman_, Dec 08 2023