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A322151
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Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).
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6
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1, 2, 5, 27, 216, 2311, 30988, 499919, 9431026, 203743252, 4960335470, 134382267082, 4009794148101, 130668970606412, 4617468180528235, 175867725701333896, 7182126650899080024, 313063334893103361130, 14507460736615554141354, 712192629608088061633746
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OFFSET
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0,2
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LINKS
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MATHEMATICA
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multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[multsubs[Range[n+1], 2], {n}], And[Union@@#==Range[Max@@Union@@#], Length[csm[#]]==1]&]], {n, 5}]
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PROG
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(PARI)
Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j+1, 2)))))} \\ Andrew Howroyd, Nov 28 2018
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CROSSREFS
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Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A291842 (planar case), A321254, A322114, A322115.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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