cp's OEIS Frontend

A323818 Number of connected set-systems covering n vertices.

%I A323818 #13 Oct 05 2022 03:24:39
%S A323818 1,1,4,96,31840,2147156736,9223372011084915712,
%T A323818 170141183460469231602560095199828453376,
%U A323818 57896044618658097711785492504343953923912733397452774312021795134847892828160
%N A323818 Number of connected set-systems covering n vertices.
%C A323818 Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.
%H A323818 G. C. Greubel, <a href="/A323818/b323818.txt">Table of n, a(n) for n = 0..11</a>
%F A323818 E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).
%F A323818 Logarithmic transform of A003465.
%e A323818 The a(2) = 4 set-systems:
%e A323818   {{1, 2}}
%e A323818   {{1}, {1,2}}
%e A323818   {{2}, {1,2}}
%e A323818   {{1}, {2}, {1,2}}
%p A323818 b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=0..n):
%p A323818 a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
%p A323818        k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
%p A323818     end:
%p A323818 seq(a(n), n=0..8);  # _Alois P. Heinz_, Jan 30 2019
%t A323818 nn=8;
%t A323818 ser=Sum[2^(2^n-1)*x^n/n!,{n,0,nn}];
%t A323818 Table[SeriesCoefficient[1-x+Log[ser],{x,0,n}]*n!,{n,0,nn}]
%o A323818 (Magma)
%o A323818 m:=12;
%o A323818 f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;
%o A323818 R<x>:=PowerSeriesRing(Rationals(), m);
%o A323818 Coefficients(R!(Laplace( f(x) ))); // _G. C. Greubel_, Oct 04 2022
%o A323818 (SageMath)
%o A323818 m=12;
%o A323818 def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))
%o A323818 def A_list(prec):
%o A323818     P.<x> = PowerSeriesRing(QQ, prec)
%o A323818     return P( f(x) ).egf_to_ogf().list()
%o A323818 A_list(m) # _G. C. Greubel_, Oct 04 2022
%Y A323818 Cf. A001187, A003465 (not necessarily connected), A016031, A048143, A092918, A293510, A317672, A323816, A323817 (no singletons), A323819 (unlabeled case).
%K A323818 nonn
%O A323818 0,3
%A A323818 _Gus Wiseman_, Jan 30 2019