cp's OEIS Frontend

A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

%I A327362 #10 Sep 11 2019 16:47:05
%S A327362 0,0,1,3,28,475,14646,813813,82060392,15251272983,5312295240010,
%T A327362 3519126783483377,4487168285715524124,11116496280631563128723,
%U A327362 53887232400918561791887118,513757147287101157620965656285,9668878162669182924093580075565776
%N A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).
%C A327362 A graph is covering if the vertex set is the union of the edge set, so there are no isolated vertices.
%H A327362 Andrew Howroyd, <a href="/A327362/b327362.txt">Table of n, a(n) for n = 0..50</a>
%H A327362 Gus Wiseman, <a href="/A327362/a327362.png">The a(4) = 28 connected covering graphs with at least one endpoint.</a>
%F A327362 Inverse binomial transform of A327364.
%F A327362 a(n) = A001187(n) - A059166(n). - _Andrew Howroyd_, Sep 11 2019
%t A327362 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327362 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]
%o A327362 (PARI) seq(n)={Vec(serlaplace(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))), -(n+1))} \\ _Andrew Howroyd_, Sep 11 2019
%Y A327362 The non-connected version is A327227.
%Y A327362 The non-covering version is A327364.
%Y A327362 Graphs with endpoints are A245797.
%Y A327362 Connected covering graphs are A001187.
%Y A327362 Connected graphs with bridges are A327071.
%Y A327362 Cf. A004110, A059166, A006125, A006129, A059166, A100743, A141580, A322395, A327105, A327229, A327230, A327366, A327369, A327377.
%K A327362 nonn
%O A327362 0,4
%A A327362 _Gus Wiseman_, Sep 04 2019
%E A327362 Terms a(7) and beyond from _Andrew Howroyd_, Sep 11 2019