cp's OEIS Frontend

A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

%I A370317 #18 Mar 29 2025 07:09:31
%S A370317 1,0,0,1,15,252,4905,110715,2864148,83838720,2744568522,99463408335,
%T A370317 3955626143040,171344363805582,8031863998136355,405150528051451000,
%U A370317 21884686370917378050,1260420510502767861840,77105349570138633021624,4993117552678619556356085
%N A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.
%H A370317 Andrew Howroyd, <a href="/A370317/b370317.txt">Table of n, a(n) for n = 0..100</a>
%F A370317 a(n) = n!*[x^n][y^n] exp(x)*(1 + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - _Andrew Howroyd_, Feb 19 2024
%e A370317 The a(0) = 0 through a(4) = 15 graphs:
%e A370317   {}  .  .  {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{1,4},{2,3}}
%e A370317                                  {{1,2},{1,3},{1,4},{2,4}}
%e A370317                                  {{1,2},{1,3},{1,4},{3,4}}
%e A370317                                  {{1,2},{1,3},{2,3},{2,4}}
%e A370317                                  {{1,2},{1,3},{2,3},{3,4}}
%e A370317                                  {{1,2},{1,3},{2,4},{3,4}}
%e A370317                                  {{1,2},{1,4},{2,3},{2,4}}
%e A370317                                  {{1,2},{1,4},{2,3},{3,4}}
%e A370317                                  {{1,2},{1,4},{2,4},{3,4}}
%e A370317                                  {{1,2},{2,3},{2,4},{3,4}}
%e A370317                                  {{1,3},{1,4},{2,3},{2,4}}
%e A370317                                  {{1,3},{1,4},{2,3},{3,4}}
%e A370317                                  {{1,3},{1,4},{2,4},{3,4}}
%e A370317                                  {{1,3},{2,3},{2,4},{3,4}}
%e A370317                                  {{1,4},{2,3},{2,4},{3,4}}
%t A370317 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 A370317 Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==n&&Length[csm[#]]<=1&]], {n,0,5}]
%o A370317 (PARI) a(n)=n!*polcoef(polcoef(exp(x + O(x*x^n))*(1 + log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k,2)*x^k/k!, O(x*x^n)))), n), n) \\ _Andrew Howroyd_, Feb 19 2024
%Y A370317 The covering case is A057500.
%Y A370317 This is the connected case of A116508.
%Y A370317 Allowing any number of edges gives A287689.
%Y A370317 Counting only covered vertices gives A370318.
%Y A370317 A006125 counts graphs, unlabeled A000088.
%Y A370317 A006129 counts covering graphs, connected A001187.
%Y A370317 A369192 counts graphs with at most n edges, covering A369191.
%Y A370317 Cf. A000169, A001862, A054780, A062740, A129271, A367863, A369193, A369197.
%K A370317 nonn
%O A370317 0,5
%A A370317 _Gus Wiseman_, Feb 17 2024
%E A370317 a(8) onwards from _Andrew Howroyd_, Feb 19 2024