A001434 - OEIS

%I M1653 N0647 #42 Feb 26 2024 09:17:49

%S 1,0,0,1,2,6,21,65,221,771,2769,10250,39243,154658,628635,2632420,

%T 11353457,50411413,230341716,1082481189,5228952960,25945377057,

%U 132140242356,690238318754,3694876952577,20252697246580,113578669178222,651178533855913,3813856010041981

%N Number of graphs with n nodes and n edges.

%C The labeled version is A116508. - _Gus Wiseman_, Feb 22 2024

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A001434/b001434.txt">Table of n, a(n) for n = 0..50</a> (terms 1..40 from Sean A. Irvine)

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/nauty">Generate graphs</a>

%H M. L. Stein and P. R. Stein, <a href="http://dx.doi.org/10.2172/4180737">Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points</a>. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.

%e From _Gus Wiseman_, Feb 22 2024: (Start)

%e Representatives of the a(0) = 1 through a(5) = 6 graphs:

%e   {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}

%e                         {12,13,24,34}  {12,13,14,23,24}

%e                                        {12,13,14,23,25}

%e                                        {12,13,14,23,45}

%e                                        {12,13,14,25,35}

%e                                        {12,13,24,35,45}

%e (End)

%t (* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfGraphs[n, n], {n, 24}] (* _Robert G. Wilson v_, Mar 22 2011 *)

%t 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 Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{2}],{n}]]],{n,0,5}] (* _Gus Wiseman_, Feb 22 2024 *)

%o (PARI) a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A008406. - _Andrew Howroyd_, Feb 02 2024

%Y The connected case is A001429, labeled A057500.

%Y The covering case is A006649, labeled A367863.

%Y Diagonal n = k of A008406.

%Y The labeled version is A116508.

%Y The version with loops is A368598, connected A368983.

%Y Allowing up to n edges gives A370315, labeled A369192.

%Y A000088 counts unlabeled graphs, labeled A006125.

%Y A001349 counts unlabeled connected graphs, labeled A001187.

%Y A002494 counts unlabeled covering graphs, labeled A006129.

%Y 
Cf. A000055, A005703, A014068, A054924, A137916, A137917, A236570, A369201.

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Jan 07 2000

%E a(0)=1 prepended by _Andrew Howroyd_, Feb 02 2024