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A323817 Number of connected set-systems covering n vertices with no singletons.

%I A323817 #10 Oct 06 2022 08:17:20
%S A323817 1,0,1,12,1990,67098648,144115187673201808,
%T A323817 1329227995784915871895000743748659792,
%U A323817 226156424291633194186662080095093570015284114833799899656335137245499581360
%N A323817 Number of connected set-systems covering n vertices with no singletons.
%H A323817 G. C. Greubel, <a href="/A323817/b323817.txt">Table of n, a(n) for n = 0..11</a>
%F A323817 Logarithmic transform of A323816.
%e A323817 The a(3) = 12 set-systems:
%e A323817   {{1, 2, 3}}
%e A323817   {{1, 2}, {1, 3}}
%e A323817   {{1, 2}, {2, 3}}
%e A323817   {{1, 3}, {2, 3}}
%e A323817   {{1, 2}, {1, 2, 3}}
%e A323817   {{1, 3}, {1, 2, 3}}
%e A323817   {{2, 3}, {1, 2, 3}}
%e A323817   {{1, 2}, {1, 3}, {2, 3}}
%e A323817   {{1, 2}, {1, 3}, {1, 2, 3}}
%e A323817   {{1, 2}, {2, 3}, {1, 2, 3}}
%e A323817   {{1, 3}, {2, 3}, {1, 2, 3}}
%e A323817   {{1, 2}, {1, 3}, {2, 3},{1, 2, 3}}
%e A323817 The A323816(4) - a(4) = 3 disconnected set-systems covering n vertices with no singletons:
%e A323817   {{1, 2}, {3, 4}}
%e A323817   {{1, 3}, {2, 4}}
%e A323817   {{1, 4}, {2, 3}}
%p A323817 b:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
%p A323817 a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
%p A323817        k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
%p A323817     end:
%p A323817 seq(a(n), n=0..8);  # _Alois P. Heinz_, Jan 30 2019
%t A323817 nn=10;
%t A323817 ser=Sum[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}]*x^n/n!,{n,0,nn}];
%t A323817 Table[SeriesCoefficient[1+Log[ser],{x,0,n}]*n!,{n,0,nn}]
%o A323817 (Magma)
%o A323817 m:=10;
%o A323817 A323816:= func< n | (&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]) >;
%o A323817 f:= func< x | 1 + Log((&+[A323816(j)*x^j/Factorial(j): j in [0..m+2]])) >;
%o A323817 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A323817 Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Oct 05 2022
%o A323817 (SageMath)
%o A323817 m=10
%o A323817 def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
%o A323817 def A323817_list(prec):
%o A323817     P.<x> = PowerSeriesRing(QQ, prec)
%o A323817     return P( 1 + log(sum(A323816(j)*x^j/factorial(j) for j in range(m+2))) ).egf_to_ogf().list()
%o A323817 A323817_list(m) # _G. C. Greubel_, Oct 05 2022
%Y A323817 Cf. A001187, A016031, A048143, A092918, A293510, A317795, A323816 (not necessarily connected), A323818 (with singletons), A323819, A323820 (unlabeled case).
%K A323817 nonn
%O A323817 0,4
%A A323817 _Gus Wiseman_, Jan 30 2019