A370315 Number of unlabeled simple graphs with n possibly isolated vertices and up to n edges.

1, 1, 2, 4, 9, 20, 54, 146, 436, 1372, 4577, 15971, 58376, 221876, 876012, 3583099, 15159817, 66248609, 298678064, 1387677971, 6637246978, 32648574416, 165002122350, 855937433641, 4553114299140, 24813471826280, 138417885372373, 789683693019999, 4603838061688077

Offset: 0

Author

Gus Wiseman, Feb 18 2024

Keywords

• nonn

Examples

			The a(1) = 1 through a(4) = 9 graph edge sets:
  {}  {}    {}          {}
      {12}  {12}        {12}
            {12-13}     {12-13}
            {12-13-23}  {12-34}
                        {12-13-14}
                        {12-13-23}
                        {12-13-24}
                        {12-13-14-23}
                        {12-13-24-34}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

The case of exactly n edges is A001434, covering A006649.

The connected covering case is A005703, labeled A129271.

Partial row sums of A008406, covering A370167.

The labeled version is A369192.

The version with loops is A370168, labeled A066383.

The covering case is A370316, labeled A369191.

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

Cf. A000666, A322661, A322700, A369194, A369196, A369197, A369199, A370169.

Programs

Mathematica

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]], Length[#]<=n&]]],{n,0,5}]

PARI

a(n) = if(n<=1, n>=0, polcoef(G(n, O(x*x^n))/(1-x),n)) \\ G(n) defined in A008406. - Andrew Howroyd, Feb 20 2024

Formula

Sum of first n+1 terms of row n of A008406.