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A327073
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Number of labeled simple connected graphs with n vertices and exactly one bridge.
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8
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0, 0, 1, 0, 12, 200, 7680, 506856, 58934848, 12205506096, 4595039095680, 3210660115278000, 4240401342141499392, 10743530775519296581944, 52808688280248604235191296, 507730995579614277599205009240, 9603347831901155679455061048606720, 358743609478638769812094362544644831968
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OFFSET
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0,5
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COMMENTS
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A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 12 graphs with exactly one bridge.
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FORMULA
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E.g.f.: (x + Sum_{k>=2} A095983(k)*x^k/(k-1)!)^2/2. - Andrew Howroyd, Aug 25 2019
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MATHEMATICA
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csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#, i]]<n||Length[csm[Delete[#, i]]]>1, {i, Length[#]}], True]==1&]], {n, 0, 5}]
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PROG
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(PARI) \\ See A095983
seq(n)={my(r=x+O(x^n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n-1, my(c=polcoeff(p, k)); r+=c*x^k; p-=c*q^k);
Vec(serlaplace(r^2/2), -(n+1))} \\ Andrew Howroyd, Aug 25 2019
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CROSSREFS
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Column k = 1 of A327072.
The unlabeled version is A327074.
Connected graphs with no bridges are A007146.
Connected graphs whose bridges are all leaves are A322395.
Connected graphs with at least one bridge are A327071.
Cf. A001187, A006129, A052446, A095983, A327069, A327077, A327108, A327111, A327145, A327146.
Sequence in context: A036240 A292056 A277311 * A133242 A141836 A083932
Adjacent sequences: A327070 A327071 A327072 * A327074 A327075 A327076
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Aug 24 2019
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EXTENSIONS
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Terms a(6) and beyond from Andrew Howroyd, Aug 25 2019
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STATUS
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approved
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