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A198046 Exponential transform of A053549.

%I A198046 #15 Jun 19 2018 05:06:40
%S A198046 1,1,3,19,225,4841,192355,14537643,2135997537,616565334097,
%T A198046 351243585487331,395958973398105283,885030941975862363649,
%U A198046 3928075680727698371316537,34658158001445631936261356547,608435501761943981290097259909211
%N A198046 Exponential transform of A053549.
%C A198046 a(n) is the number of ways to designate a node in each connected component over all simple labeled graphs on n nodes.
%H A198046 Alois P. Heinz, <a href="/A198046/b198046.txt">Table of n, a(n) for n = 0..80</a>
%F A198046 E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A053549.
%p A198046 g:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
%p A198046       add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*g(k), k=1..n-1)/n)
%p A198046     end:
%p A198046 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A198046       binomial(n-1, j-1) *j*g(j) *a(n-j), j=1..n))
%p A198046     end:
%p A198046 seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 17 2015
%t A198046 nn=20; a=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}]; Range[0,nn]! CoefficientList[Series[Exp[x D[Log[a],x]], {x,0,nn}], x]
%o A198046 (PARI) seq(n)={Vec(serlaplace(exp(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))))))} \\ _Andrew Howroyd_, Jun 18 2018
%Y A198046 Cf. A053549.
%K A198046 nonn
%O A198046 0,3
%A A198046 _Geoffrey Critzer_, Oct 20 2011
%E A198046 a(6), a(10) corrected by _Alois P. Heinz_, Mar 18 2015