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A327362
Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).
6
0, 0, 1, 3, 28, 475, 14646, 813813, 82060392, 15251272983, 5312295240010, 3519126783483377, 4487168285715524124, 11116496280631563128723, 53887232400918561791887118, 513757147287101157620965656285, 9668878162669182924093580075565776
OFFSET
0,4
COMMENTS
A graph is covering if the vertex set is the union of the edge set, so there are no isolated vertices.
FORMULA
Inverse binomial transform of A327364.
a(n) = A001187(n) - A059166(n). - Andrew Howroyd, Sep 11 2019
MATHEMATICA
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}]
PROG
(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
CROSSREFS
The non-connected version is A327227.
The non-covering version is A327364.
Graphs with endpoints are A245797.
Connected covering graphs are A001187.
Connected graphs with bridges are A327071.
Sequence in context: A210854 A060545 A108288 * A327071 A346315 A058804
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Sep 11 2019
STATUS
approved