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A323816 Number of set-systems covering n vertices with no singletons.

%I A323816 #11 Oct 06 2022 08:17:12
%S A323816 1,0,1,12,1993,67098768,144115187673233113,
%T A323816 1329227995784915871895000745158568460,
%U A323816 226156424291633194186662080095093570015284114833799899660370362545578585265
%N A323816 Number of set-systems covering n vertices with no singletons.
%H A323816 G. C. Greubel, <a href="/A323816/b323816.txt">Table of n, a(n) for n = 0..11</a>
%F A323816 Inverse binomial transform of A016031 shifted once to the left.
%e A323816 The a(3) = 12 set-systems:
%e A323816   {{1,2,3}}
%e A323816   {{1,2}, {1,3}}
%e A323816   {{1,2}, {2,3}}
%e A323816   {{1,3}, {2,3}}
%e A323816   {{1,2}, {1,2,3}}
%e A323816   {{1,3}, {1,2,3}}
%e A323816   {{2,3}, {1,2,3}}
%e A323816   {{1,2}, {1,3}, {2,3}}
%e A323816   {{1,2}, {1,3}, {1,2,3}}
%e A323816   {{1,2}, {2,3}, {1,2,3}}
%e A323816   {{1,3}, {2,3}, {1,2,3}}
%e A323816   {{1,2}, {1,3}, {2,3}, {1,2,3}}
%p A323816 a:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
%p A323816 seq(a(n), n=0..8);  # _Alois P. Heinz_, Jan 30 2019
%t A323816 Table[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}],{n,0,8}]
%o A323816 (Magma)
%o A323816 [(&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]): n in [0..12]]; // _G. C. Greubel_, Oct 05 2022
%o A323816 (SageMath)
%o A323816 def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
%o A323816 [A323816(n) for n in range(12)] # _G. C. Greubel_, Oct 05 2022
%Y A323816 Cf. A000295, A000371, A000612, A003465 (with singletons), A006129 (covers by pairs), A016031, A055154, A055621, A305001, A317795 (unlabeled case), A323817 (connected case).
%K A323816 nonn
%O A323816 0,4
%A A323816 _Gus Wiseman_, Jan 30 2019