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A000664 Number of graphs with n edges.

%I A000664 M1400 N0545 #68 Jan 20 2025 12:37:06
%S A000664 1,1,2,5,11,26,68,177,497,1476,4613,15216,52944,193367,740226,2960520,
%T A000664 12334829,53394755,239544624,1111261697,5320103252,26237509076,
%U A000664 133087001869,693339241737,3705135967663,20286965943329,113694201046379,651571521170323,3815204365835840,22806847476040913,139088381010541237,864777487052916454
%N A000664 Number of graphs with n edges.
%C A000664 These are simple graphs, unlabeled, with no isolated nodes, but are not necessarily connected.
%D A000664 W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
%D A000664 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
%D A000664 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000664 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000664 Max Alekseyev, <a href="/A000664/b000664.txt">Table of n, a(n) for n = 0..60</a>
%H A000664 Nicolas Borie, <a href="http://arxiv.org/abs/1511.05843">The Hopf Algebra of graph invariants</a>, arXiv preprint arXiv:1511.05843 [math.CO], 2015.
%H A000664 P. J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000664 Fran Herr and Legrand Jones II, <a href="https://arxiv.org/abs/2205.01796">Iterated Jump Graphs</a>, arXiv:2205.01796 [math.CO], 2022.
%H A000664 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
%H A000664 Peter Steinbach, <a href="/A000664/a000664.txt">Field Guide to Simple Graphs, Volume 4</a>, Overview of the 11 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A000664 Peter Steinbach, <a href="/A000664/a000664_1.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 1
%H A000664 Peter Steinbach, <a href="/A000664/a000664_2.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 2
%H A000664 Peter Steinbach, <a href="/A000664/a000664_3.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 3
%H A000664 Peter Steinbach, <a href="/A000664/a000664_4.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 4
%H A000664 Peter Steinbach, <a href="/A000664/a000664_5.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 5
%H A000664 Peter Steinbach, <a href="/A000664/a000664_6.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 6
%H A000664 Peter Steinbach, <a href="/A000664/a000664_7.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 7
%H A000664 Peter Steinbach, <a href="/A000664/a000664_8.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 8
%H A000664 Peter Steinbach, <a href="/A000664/a000664_9.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 9
%H A000664 Peter Steinbach, <a href="/A000664/a000664_10.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 10
%H A000664 Peter Steinbach, <a href="/A000664/a000664_11.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 11
%F A000664 a(n) = A008406(2*n,n). - _Max Alekseyev_, Sep 13 2016
%F A000664 Euler transform of A002905 (ignoring A002905(0)). - _Franklin T. Adams-Watters_ Jul 03 2009
%e A000664 n=1: o-o (1)
%e A000664 n=2: o-o o-o, o-o-o (2)
%e A000664 n=3: o-o o-o o-o, o-o-o o-o, o-o-o-o, Y, triangle (5)
%e A000664 n=4: o-o o-o o-o o-o, o-o-o o-o o-o, o-o-o o-o-o, o-o o-o-o-o, o-o Y, o-o triangle,
%e A000664 o-o-o-o-o, >o-o-o, ><, square, triangle with tail (11)
%t A000664 << Combinatorica`; Table[NumberOfGraphs[2 n, n], {n, 0, 10}] (* _Eric W. Weisstein_, Oct 30 2017 *)
%t A000664 << Combinatorica`; Table[Coefficient[GraphPolynomial[2 n, x], x, n], {n, 0, 10}] (* _Eric W. Weisstein_, Oct 30 2017 *)
%Y A000664 Cf. A002905, A008406, A053418.
%Y A000664 Row sums of A275421.
%Y A000664 Cf. also A000088, A000055.
%K A000664 nonn,nice
%O A000664 0,3
%A A000664 _N. J. A. Sloane_
%E A000664 More terms from _Vladeta Jovovic_, Jan 08 2000, Aug 14 2007
%E A000664 Edited by _N. J. A. Sloane_, Feb 26 2008
%E A000664 Example for n=2 corrected by Adrian Falcone (falcone(AT)gmail.com), Jan 28 2009
%E A000664 Zeroth term inserted by _Franklin T. Adams-Watters_, Jul 03 2009
%E A000664 a(25)-a(26) from _Max Alekseyev_, Sep 19 2009
%E A000664 a(27)-a(60) from _Max Alekseyev_, Sep 07 2016