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A095983 Number of 2-edge-connected labeled graphs on n nodes.

%I A095983 #57 Jul 23 2023 17:06:20
%S A095983 0,0,0,1,10,253,11968,1047613,169181040,51017714393,29180467201536,
%T A095983 32121680070545657,68867078000231169536,290155435185687263172693,
%U A095983 2417761175748567327193407488,40013922635723692336670167608181,1318910073755307133701940625759574016
%N A095983 Number of 2-edge-connected labeled graphs on n nodes.
%C A095983 From _Falk Hüffner_, Jun 28 2018: (Start)
%C A095983 Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
%C A095983 Labeled version of A007146. (End)
%C A095983 The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - _Gus Wiseman_, Sep 20 2019
%H A095983 Jinyuan Wang, <a href="/A095983/b095983.txt">Table of n, a(n) for n = 0..82</a>
%H A095983 P. Hanlon and R. W. Robinson, <a href="http://dx.doi.org/10.1016/0095-8956(82)90048-X">Counting bridgeless graphs</a>, J. Combin. Theory, B 33 (1982), 276-305.
%H A095983 Gus Wiseman, <a href="/A095983/a095983.png">The a(4) = 10 labeled graphs with spanning edge-connectivity >= 2.</a>
%H A095983 Gus Wiseman, <a href="/A095983/a095983_1.png">Non-isomorphic representatives of the a(5) = 253 graphs with spanning edge-connectivity >= 2.</a>
%F A095983 a(n) = A001187(n) - A327071(n). - _Gus Wiseman_, Sep 20 2019
%t A095983 seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
%t A095983 seq[16] (* _Jean-François Alcover_, Aug 13 2019, after _Andrew Howroyd_ *)
%t A095983 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 A095983 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t A095983 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]>=2&]],{n,0,5}] (* _Gus Wiseman_, Sep 20 2019 *)
%o A095983 (PARI) \\ here p is initially A053549, q is A198046 as e.g.f.s.
%o A095983 seq(n)={my(v=vector(n));
%o A095983 my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
%o A095983 my(q=x*exp(p)); p-=q;
%o A095983 for(k=3, n, my(c=polcoeff(p,k)); v[k]=c*(k-1)!; p-=c*q^k);
%o A095983 concat([0],v)} \\ _Andrew Howroyd_, Jun 18 2018
%o A095983 (PARI) seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ _Andrew Howroyd_, Dec 28 2020
%Y A095983 Cf. A053549, A198046.
%Y A095983 The unlabeled version is A007146.
%Y A095983 Row sums of A327069 if the first two columns are removed.
%Y A095983 BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
%Y A095983 Graphs with spanning edge-connectivity 2 are A327146.
%Y A095983 Graphs with non-spanning edge-connectivity >= 2 are A327200.
%Y A095983 2-vertex-connected graphs are A013922.
%Y A095983 Graphs without endpoints are A059167.
%Y A095983 Graphs with spanning edge-connectivity 1 are A327071.
%Y A095983 Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.
%K A095983 nonn
%O A095983 0,5
%A A095983 Yifei Chen (yifei(AT)mit.edu), Jul 17 2004
%E A095983 Name corrected and more terms from _Pavel Irzhavski_, Nov 01 2014
%E A095983 Offset corrected by _Falk Hüffner_, Jun 17 2018
%E A095983 a(12)-a(16) from _Andrew Howroyd_, Jun 18 2018