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A370165 Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

%I A370165 #16 Feb 20 2024 21:52:38
%S A370165 1,1,4,29,400,10289,496548,45455677,7983420736,2716094133313,
%T A370165 1803251169342820,2348787270663723581,6024912118926389490448,
%U A370165 30516957491540079828757553,305811332460677494410532494660,6071677788061208810793717466942237
%N A370165 Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.
%C A370165 Number of ways to choose a stable vertex set of a simple graph with n vertices.
%H A370165 Andrew Howroyd, <a href="/A370165/b370165.txt">Table of n, a(n) for n = 0..80</a>
%F A370165 Inverse binomial transform of A079491.
%F A370165 E.g.f.: Sum_{k >= 0} exp((2^k-1)*x)*2^(k*(k-1)/2)*x^k/k!. - _Andrew Howroyd_, Feb 20 2024
%e A370165 The a(3) = 29 loop-graphs (loops shown as singletons):
%e A370165   {1,23}   {1,2,3}     {1,2,13,23}
%e A370165   {2,13}   {1,2,13}    {1,3,12,23}
%e A370165   {3,12}   {1,2,23}    {2,3,12,13}
%e A370165   {12,13}  {1,3,12}    {1,12,13,23}
%e A370165   {12,23}  {1,3,23}    {2,12,13,23}
%e A370165   {13,23}  {2,3,12}    {3,12,13,23}
%e A370165            {2,3,13}
%e A370165            {1,12,13}
%e A370165            {1,12,23}
%e A370165            {1,13,23}
%e A370165            {2,12,13}
%e A370165            {2,12,23}
%e A370165            {2,13,23}
%e A370165            {3,12,13}
%e A370165            {3,12,23}
%e A370165            {3,13,23}
%e A370165            {12,13,23}
%t A370165 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&!MatchQ[#, {___,{x_},___,{y_},___,{x_,y_},___}]&]],{n,0,5}]
%o A370165 (PARI) seq(n)={Vec(serlaplace(sum(k=0, n, exp((2^k-1)*x + O(x*x^n))*2^(k*(k-1)/2)*x^k/k!)))} \\ _Andrew Howroyd_, Feb 20 2024
%Y A370165 Without loops we have A006129, connected A001187.
%Y A370165 The non-covering version is A079491.
%Y A370165 The unlabeled version is A370166, non-covering A339832.
%Y A370165 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A370165 A000666 counts unlabeled loop-graphs, covering A322700.
%Y A370165 A006125 counts labeled loop-graphs (shifted left), covering A322661.
%Y A370165 Cf. A048291, A062740, A116539, A320461, A322635.
%K A370165 nonn
%O A370165 0,3
%A A370165 _Gus Wiseman_, Feb 12 2024