A370318 - OEIS

%I #14 Feb 20 2024 02:29:17

%S 0,0,0,1,19,307,5237,99137,2098946,49504458,1291570014,37002273654,

%T 1156078150969,39147186978685,1428799530304243,55933568895261791,

%U 2338378885159906196,103995520598384132516,4903038902046860966220,244294315694676224001852,12827355456239840407125363

%N Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

%C The case of an empty edge set is excluded.

%H Andrew Howroyd, <a href="/A370318/b370318.txt">Table of n, a(n) for n = 0..100</a>

%F Binomial transform of A057500 (if the null graph is not connected).

%F a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!). - _Andrew Howroyd_, Feb 19 2024

%t 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 Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]

%o (PARI) \\ Compare A370317; use A057500 for efficiency.

%o a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+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 The covering case is A057500, which is also the covering case of A370317.

%Y This is the connected case of A367862, covering A367863.

%Y A001187 counts connected graphs, A001349 unlabeled.

%Y A006125 counts graphs, A000088 unlabeled.

%Y A006129 counts covering graphs, A002494 unlabeled.

%Y A062734 counts connected graphs by edge count.

%Y A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.

%Y A143543 counts simple labeled graphs by number of connected components.

%Y A367867 = graphs contradict strict AoC, connected A140638, covering A367868.

%Y Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

%K nonn

%O 0,5

%A _Gus Wiseman_, Feb 18 2024