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%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
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