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Number of connected set-systems covering n vertices.
111

%I #13 Oct 05 2022 03:24:39

%S 1,1,4,96,31840,2147156736,9223372011084915712,

%T 170141183460469231602560095199828453376,

%U 57896044618658097711785492504343953923912733397452774312021795134847892828160

%N Number of connected set-systems covering n vertices.

%C Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.

%H G. C. Greubel, <a href="/A323818/b323818.txt">Table of n, a(n) for n = 0..11</a>

%F E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).

%F Logarithmic transform of A003465.

%e The a(2) = 4 set-systems:

%e {{1, 2}}

%e {{1}, {1,2}}

%e {{2}, {1,2}}

%e {{1}, {2}, {1,2}}

%p b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=0..n):

%p a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(

%p k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)

%p end:

%p seq(a(n), n=0..8); # _Alois P. Heinz_, Jan 30 2019

%t nn=8;

%t ser=Sum[2^(2^n-1)*x^n/n!,{n,0,nn}];

%t Table[SeriesCoefficient[1-x+Log[ser],{x,0,n}]*n!,{n,0,nn}]

%o (Magma)

%o m:=12;

%o f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Oct 04 2022

%o (SageMath)

%o m=12;

%o def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))

%o def A_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(x) ).egf_to_ogf().list()

%o A_list(m) # _G. C. Greubel_, Oct 04 2022

%Y Cf. A001187, A003465 (not necessarily connected), A016031, A048143, A092918, A293510, A317672, A323816, A323817 (no singletons), A323819 (unlabeled case).

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 30 2019