cp's OEIS Frontend

A327364 Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).

%I A327364 #8 Sep 11 2019 16:46:57
%S A327364 0,0,1,6,46,655,17991,927416,89009740,16020407709,5468601546685,
%T A327364 3578414666656214,4529751815161579194,11175105490563109463875,
%U A327364 54043272967471942825421219,514566625051705610110588073460,9677104749727084630538798805505880
%N A327364 Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).
%H A327364 Andrew Howroyd, <a href="/A327364/b327364.txt">Table of n, a(n) for n = 0..50</a>
%F A327364 Binomial transform of A327362.
%e A327364 The a(4) = 46 edge-sets:
%e A327364   {12}  {12,13}  {12,13,14}  {12,13,14,23}
%e A327364   {13}  {12,14}  {12,13,24}  {12,13,14,24}
%e A327364   {14}  {12,23}  {12,13,34}  {12,13,14,34}
%e A327364   {23}  {12,24}  {12,14,23}  {12,13,23,24}
%e A327364   {24}  {13,14}  {12,14,34}  {12,13,23,34}
%e A327364   {34}  {13,23}  {12,23,24}  {12,14,23,24}
%e A327364         {13,34}  {12,23,34}  {12,14,24,34}
%e A327364         {14,24}  {12,24,34}  {12,23,24,34}
%e A327364         {14,34}  {13,14,23}  {13,14,23,34}
%e A327364         {23,24}  {13,14,24}  {13,14,24,34}
%e A327364         {23,34}  {13,23,24}  {13,23,24,34}
%e A327364         {24,34}  {13,23,34}  {14,23,24,34}
%e A327364                  {13,24,34}
%e A327364                  {14,23,24}
%e A327364                  {14,23,34}
%e A327364                  {14,24,34}
%t A327364 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 A327364 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]
%o A327364 (PARI) seq(n)={my(x=x + O(x*x^n)); Vec(serlaplace(exp(x)*(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!)) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x))^k/k!)))), -(n+1))} \\ _Andrew Howroyd_, Sep 11 2019
%Y A327364 The covering case is A327362.
%Y A327364 Graphs with endpoints are A245797.
%Y A327364 Graphs with connected edge-set are A287689.
%Y A327364 Connected graphs with bridges are A327071.
%Y A327364 Covering graphs with endpoints are A327227.
%Y A327364 Cf. A001187, A059167, A141580, A322395, A327148, A327335, A327369.
%K A327364 nonn
%O A327364 0,4
%A A327364 _Gus Wiseman_, Sep 04 2019
%E A327364 Terms a(7) and beyond from _Andrew Howroyd_, Sep 11 2019