cp's OEIS Frontend

A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.

%I A370316 #10 Feb 19 2024 22:56:32
%S A370316 1,0,1,2,5,10,28,68,193,534,1568,4635,14146,43610,137015,435227,
%T A370316 1400058,4547768,14917504,49348612,164596939,553177992,1872805144,
%U A370316 6385039022,21917878860,75739158828,263438869515,922219844982,3249042441125,11519128834499,41097058489426
%N A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.
%H A370316 Andrew Howroyd, <a href="/A370316/b370316.txt">Table of n, a(n) for n = 0..50</a>
%e A370316 The a(0) = 1 through a(5) = 10 simple graphs:
%e A370316   {}  .  {12}  {12-13}     {12-34}        {12-13-45}
%e A370316                {12-13-23}  {12-13-14}     {12-13-14-15}
%e A370316                            {12-13-24}     {12-13-14-25}
%e A370316                            {12-13-14-23}  {12-13-23-45}
%e A370316                            {12-13-24-34}  {12-13-24-35}
%e A370316                                           {12-13-14-15-23}
%e A370316                                           {12-13-14-23-25}
%e A370316                                           {12-13-14-23-45}
%e A370316                                           {12-13-14-25-35}
%e A370316                                           {12-13-24-35-45}
%t A370316 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
%t A370316 Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}],{0,n}], Union@@#==Range[n]&]]],{n,0,5}]
%o A370316 (PARI) \\ G defined in A008406.
%o A370316 a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ _Andrew Howroyd_, Feb 19 2024
%Y A370316 The connected case is A005703, labeled A129271.
%Y A370316 The case of exactly n edges is A006649, covering case of A001434.
%Y A370316 The labeled version is A369191.
%Y A370316 Partial row sums of A370167, covering case of A008406.
%Y A370316 The non-covering version with loops is A370168, labeled A066383.
%Y A370316 The version with loops is A370169, labeled A369194.
%Y A370316 The non-covering version is A370315, labeled A369192.
%Y A370316 A006125 counts graphs, unlabeled A000088.
%Y A370316 A006129 counts covering graphs, unlabeled A002494.
%Y A370316 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A370316 Cf. A054548, A062734, A116508, A367862, A367863, A368599, A369193.
%K A370316 nonn
%O A370316 0,4
%A A370316 _Gus Wiseman_, Feb 18 2024
%E A370316 a(8) onwards from _Andrew Howroyd_, Feb 19 2024