This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000088 M1253 N0479 #361 Feb 16 2025 08:32:19
%S A000088 1,1,2,4,11,34,156,1044,12346,274668,12005168,1018997864,165091172592,
%T A000088 50502031367952,29054155657235488,31426485969804308768,
%U A000088 64001015704527557894928,245935864153532932683719776,1787577725145611700547878190848,24637809253125004524383007491432768
%N A000088 Number of simple graphs on n unlabeled nodes.
%C A000088 Euler transform of the sequence A001349.
%C A000088 Also, number of equivalence classes of sign patterns of totally nonzero symmetric n X n matrices.
%D A000088 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 430.
%D A000088 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 519.
%D A000088 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 214.
%D A000088 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 240.
%D A000088 Thomas Boyer-Kassem, Conor Mayo-Wilson, Scientific Collaboration and Collective Knowledge: New Essays, New York, Oxford University Press, 2018, see page 47.
%D A000088 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 54.
%D A000088 Lupanov, O. B. Asymptotic estimates of the number of graphs with n edges. (Russian) Dokl. Akad. Nauk SSSR 126 1959 498--500. MR0109796 (22 #681).
%D A000088 M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22.
%D A000088 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000088 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A000088 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000088 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000088 Keith M. Briggs, <a href="/A000088/b000088.txt">Table of n, a(n) for n = 0..87</a> (From link below)
%H A000088 R. Absil and H. Mélot, <a href="http://arxiv.org/abs/1304.7993">Digenes: genetic algorithms to discover conjectures about directed and undirected graphs</a>, arXiv preprint arXiv:1304.7993 [cs.DM], 2013.
%H A000088 Natalie Arkus, Vinothan N. Manoharan and Michael P. Brenner. <a href="http://arxiv.org/abs/1011.5412"> Deriving Finite Sphere Packings</a>, arXiv:1011.5412 [cond-mat.soft], Nov 24, 2010. (See Table 1.)
%H A000088 Leonid Bedratyuk and Anna Bedratyuk, <a href="http://www.proceedings.bas.bg/cgi-bin/mitko/0DOC_abs.pl?2016_3_02">A new formula for the generating function of the numbers of simple graphs</a>, Comptes rendus de l'Académie Bulgare des Sciences, Tome 69, No 3, 2016, p.259-268.
%H A000088 Benjamin A. Blumer, Michael S. Underwood and David L. Feder, <a href="http://arxiv.org/abs/1111.5032">Single-qubit unitary gates by graph scattering</a>, arXiv:1111.5032 [quant-ph], 2011.
%H A000088 Keith M. Briggs, <a href="http://keithbriggs.info/cgt.html">Combinatorial Graph Theory</a> [Gives first 140 terms]
%H A000088 Gunnar Brinkmann, Kris Coolsaet, Jan Goedgebeur and Hadrien Melot, <a href="http://arxiv.org/abs/1204.3549">House of Graphs: a database of interesting graphs</a>, arXiv preprint arXiv:1204.3549 [math.CO], 2012.
%H A000088 P. Butler and R. W. Robinson, <a href="/A002218/a002218.pdf">On the computer calculation of the number of nonseparable graphs</a>, pp. 191 - 208 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp. [Annotated scanned copy]
%H A000088 P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102.
%H A000088 P. J. Cameron, <a href="http://dx.doi.org/10.1016/S0167-5060(08)70569-7">Some sequences of integers</a>, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H A000088 P. J. Cameron, <a href="http://www.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 A000088 P. J. Cameron and C. R. Johnson, <a href="http://dx.doi.org/10.1016/j.disc.2004.10.029">The number of equivalence patterns of symmetric sign patterns</a>, Discr. Math., 306 (2006), 3074-3077.
%H A000088 Gi-Sang Cheon, Jinha Kim, Minki Kim and Sergey Kitaev, <a href="https://arxiv.org/abs/1803.01055">On k-11-representable graphs</a>, arXiv:1803.01055 [math.CO], 2018.
%H A000088 CombOS - Combinatorial Object Server, <a href="http://combos.org/nauty">Generate graphs</a>
%H A000088 Karen M. Daehli, Sarah A Obead, Hsuan-Yin Lin, and Eirik Rosnes, <a href="https://arxiv.org/abs/2401.06125">Improved Capacity Outer Bound for Private Quadratic Monomial Computation</a>, arXiv:2401.06125 [cs.IR], 2024.
%H A000088 R. L. Davis, <a href="http://dx.doi.org/10.1090/S0002-9939-1953-0055294-2">The number of structures of finite relations</a>, Proc. Amer. Math. Soc. 4 (1953), 486-495.
%H A000088 J. P. Dolch, <a href="/A001349/a001349.pdf">Names of Hamiltonian graphs</a>, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271. (Annotated scanned copy of 3 pages)
%H A000088 Peter Dukes, <a href="http://www.math.uvic.ca/courses/2019s/math422/a01/M422-notes.pdf">Notes for Math 422: Enumeration and Ramsey Theory</a>, University of Victoria BC Canada (2019). See page 36.
%H A000088 D. S. Dummit, E. P. Dummit and H. Kisilevsky, <a href="http://arxiv.org/abs/1512.06480">Characterizations of quadratic, cubic, and quartic residue matrices</a>, arXiv preprint arXiv:1512.06480 [math.NT], 2015.
%H A000088 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
%H A000088 Mareike Fischer, Michelle Galla, Lina Herbst, Yangjing Long, and Kristina Wicke, <a href="https://arxiv.org/abs/1810.06853">Non-binary treebased unrooted phylogenetic networks and their relations to binary and rooted ones</a>, arXiv:1810.06853 [q-bio.PE], 2018.
%H A000088 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, 2009; see page 105.
%H A000088 E. Friedman, <a href="/A000088/a000088a.gif">Illustration of small graphs</a>
%H A000088 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_41.html">Graphs</a>
%H A000088 Scott Garrabrant and Igor Pak, <a href="http://www.math.ucla.edu/~pak/papers/PatternAvoid10.pdf">Pattern Avoidance is Not P-Recursive</a>, preprint, 2015.
%H A000088 Lee M. Gunderson and Gecia Bravo-Hermsdorff, <a href="https://arxiv.org/abs/2002.03959">Introducing Graph Cumulants: What is the Variance of Your Social Network?</a>, arXiv:2002.03959 [math.ST], 2020.
%H A000088 P. Hegarty, <a href="http://arxiv.org/abs/1212.4303">On the notion of balance in social network analysis</a>, arXiv preprint arXiv:1212.4303 [cs.SI], 2012.
%H A000088 S. Hougardy, <a href="http://www.or.uni-bonn.de/~hougardy/">Home Page</a>
%H A000088 S. Hougardy, <a href="http://dx.doi.org/10.1016/j.disc.2006.05.021">Classes of perfect graphs</a>, Discr. Math. 306 (2006), 2529-2571.
%H A000088 Richard Hua and Michael J. Dinneen, <a href="https://doi.org/10.1007/s42979-019-0020-1">Improved QUBO Formulation of the Graph Isomorphism Problem</a>, SN Computer Science (2020) Vol. 1, No. 19.
%H A000088 A. Itzhakov and M. Codish, <a href="http://arxiv.org/abs/1511.08205">Breaking Symmetries in Graph Search with Canonizing Sets</a>, arXiv preprint arXiv:1511.08205 [cs.AI], 2015-2016.
%H A000088 Dan-Marian Joiţa and Lorentz Jäntschi, <a href="https://doi.org/10.3390/math5040084">Extending the Characteristic Polynomial for Characterization of C_20 Fullerene Congeners</a>, Mathematics (2017), 5(4), 84.
%H A000088 Vladeta Jovovic, <a href="/A063843/a063843.rtf">Formulae for the number T(n,k) of n-multigraphs on k nodes</a>
%H A000088 Maksim Karev, <a href="http://arxiv.org/abs/1404.0026">The space of framed chord diagrams as a Hopf module</a>, arXiv preprint arXiv:1404.0026 [math.GT], 2014.
%H A000088 J. M. Larson, <a href="https://www.sas.upenn.edu/polisci/sites/www.sas.upenn.edu.polisci/files/Larson_cheaters_Feb2014.pdf">Cheating Because They Can: Social Networks and Norm Violators</a>, 2014. See Footnote 11.
%H A000088 Steffen Lauritzen, Alessandro Rinaldo and Kayvan Sadeghi, <a href="https://arxiv.org/abs/1709.03885">On Exchangeability in Network Models</a>, arXiv:1709.03885 [math.ST], 2017.
%H A000088 O. B. Lupanov, <a href="http://new.math.msu.su/department/dm/dmmc/PUBL2/Lupanov-graph.pdf">On asymptotic estimates of the number of graphs and networks with n edges</a>, Problems of Cybernetics [in Russian], Moscow 4 (1960): 5-21.
%H A000088 M. D. McIlroy, <a href="/A000088/a000088.pdf">Calculation of numbers of structures of relations on finite sets</a>, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
%H A000088 B. D. McKay, <a href="http://www.csse.uwa.edu.au/~gordon/remote/graphs/graphcount">Maple program</a> (used to redirect to <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/graphs/graphcount">here</a>).
%H A000088 B. D. McKay, <a href="/A000088/a000088_1.txt">Maple program</a> [Cached copy, with permission]
%H A000088 B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/graphs.html">Simple graphs</a>
%H A000088 A. Milicevic and N. Trinajstic, <a href="http://www.sicmm.org/~FAMNIT-knjiga/wwwANG/web-pages/405_469_tcm18-67839%5B1%5D.pdf">Combinatorial Enumeration in Chemistry</a>, Chem. Modell., Vol. 4, (2006), pp. 405-469.
%H A000088 Angelo Monti and Blerina Sinaimeri, <a href="https://arxiv.org/abs/2410.16023">Effects of graph operations on star pairwise compatibility graphs</a>, arXiv:2410.16023 [cs.DM], 2024. See p. 16.
%H A000088 W. Oberschelp, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002298732">Kombinatorische Anzahlbestimmungen in Relationen</a>, Math. Ann., 174 (1967), 53-78.
%H A000088 M. Petkovsek and T. Pisanski, <a href="http://hrcak.srce.hr/file/4232">Counting disconnected structures: chemical trees, fullerenes, I-graphs and others</a>, Croatica Chem. Acta, 78 (2005), 563-567.
%H A000088 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000088 E. M. Palmer, <a href="/A000088/a000088_EMP.pdf">Letter to N. J. A. Sloane, no date</a>
%H A000088 Marko Riedel, <a href="http://www.roard.com/docs/cookbook/cbsu5.html#x8-170001.5">Nonisomorphic graphs</a>.
%H A000088 Marko Riedel, <a href="/A000088/a000088_4.maple.txt">Compact Maple code for cycle index, sequence values and ordinary generating function by the number of edges.</a>
%H A000088 R. W. Robinson, <a href="http://dx.doi.org/10.1016/S0021-9800(70)80089-8">Enumeration of non-separable graphs</a>, J. Combin. Theory 9 (1970), 327-356.
%H A000088 Sage, <a href="http://www.sagemath.org/doc/reference/graphs/sage/graphs/graph_generators.html">Common Graphs (Graph Generators)</a>
%H A000088 S. S. Skiena, <a href="http://www.cs.sunysb.edu/~algorith/files/generating-graphs.shtml">Generating graphs</a>
%H A000088 N. J. A. Sloane, <a href="/A000088/a000088.gif">Illustration of initial terms</a>
%H A000088 Quico Spaen, Christopher Thraves Caro and Mark Velednitsky, <a href="https://doi.org/10.1007/s00454-019-00114-w">The Dimension of Valid Distance Drawings of Signed Graphs</a>, Discrete & Computational Geometry (2019), 1-11.
%H A000088 P. R. Stein, <a href="/A004250/a004250.pdf">On the number of graphical partitions</a>, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
%H A000088 Peter Steinbach, <a href="/A000088/a000088.txt">Field Guide to Simple Graphs, Volume 1</a>, Overview of the 17 Parts (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A000088 Peter Steinbach, <a href="/A000088/a000088_1.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 1
%H A000088 Peter Steinbach, <a href="/A000088/a000088_2.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 2
%H A000088 Peter Steinbach, <a href="/A000088/a000088_3.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 3
%H A000088 Peter Steinbach, <a href="/A000088/a000088_4.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 4
%H A000088 Peter Steinbach, <a href="/A000088/a000088_5.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 5
%H A000088 Peter Steinbach, <a href="/A000088/a000088_6.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 6
%H A000088 Peter Steinbach, <a href="/A000088/a000088_7.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 7
%H A000088 Peter Steinbach, <a href="/A000088/a000088_8.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 8
%H A000088 Peter Steinbach, <a href="/A000088/a000088_9.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 9
%H A000088 Peter Steinbach, <a href="/A000088/a000088_10.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 10
%H A000088 Peter Steinbach, <a href="/A000088/a000088_11.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 11
%H A000088 Peter Steinbach, <a href="/A000088/a000088_12.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 12
%H A000088 Peter Steinbach, <a href="/A000088/a000088_13.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 13
%H A000088 Peter Steinbach, <a href="/A000088/a000088_14.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 14
%H A000088 Peter Steinbach, <a href="/A000088/a000088_15.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 15
%H A000088 Peter Steinbach, <a href="/A000088/a000088_16.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 16
%H A000088 Peter Steinbach, <a href="/A000088/a000088_17.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 17
%H A000088 J. M. Tangen and N. J. A. Sloane, <a href="/A000666/a000666.pdf">Correspondence, 1976-1976</a>
%H A000088 James Turner and William H. Kautz, <a href="http://dx.doi.org/10.1137/1012125">A survey of progress in graph theory in the Soviet Union</a> SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 18.
%H A000088 S. Uijlen and B. Westerbaan, <a href="http://arxiv.org/abs/1412.8544">A Kochen-Specker system has at least 22 vectors</a>, arXiv preprint arXiv:1412.8544 [cs.DM], 2014.
%H A000088 Mark Velednitsky, <a href="https://marvel2010.github.io/static/Dissertation_Velednitsky_Final.pdf">New Algorithms for Three Combinatorial Optimization Problems on Graphs</a>, Ph. D. Dissertation, University of California, Berkeley (2020).
%H A000088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%H A000088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>
%H A000088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence</a>
%H A000088 E. M. Wright, <a href="http://dx.doi.org/10.1007/BF01350794">The number of graphs on many unlabelled nodes</a>, Mathematische Annalen, December 1969, Volume 183, Issue 4, 250-253
%H A000088 E. M. Wright, <a href="http://dx.doi.org/10.1090/S0002-9904-1972-13097-0 ">The number of unlabelled graphs with many nodes and edges</a> Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 1032-1034.
%H A000088 Chris Ying, <a href="https://arxiv.org/abs/1902.06192">Enumerating Unique Computational Graphs via an Iterative Graph Invariant</a>, arXiv:1902.06192 [cs.DM], 2019.
%H A000088 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A000088 a(n) = 2^binomial(n, 2)/n!*(1+(n^2-n)/2^(n-1)+8*n!/(n-4)!*(3*n-7)*(3*n-9)/2^(2*n)+O(n^5/2^(5*n/2))) (see Harary, Palmer reference). - _Vladeta Jovovic_ and _Benoit Cloitre_, Feb 01 2003
%F A000088 a(n) = 2^binomial(n, 2)/n!*[1+2*n$2*2^{-n}+8/3*n$3*(3n-7)*2^{-2n}+64/3*n$4*(4n^2-34n+75)*2^{-3n}+O(n^8*2^{-4*n})] where n$k is the falling factorial: n$k = n(n-1)(n-2)...(n-k+1). - _Keith Briggs_, Oct 24 2005
%F A000088 From David Pasino (davepasino(AT)yahoo.com), Jan 31 2009: (Start)
%F A000088 a(n) = a(n, 2), where a(n, t) is the number of t-uniform hypergraphs on n unlabeled nodes (cf. A000665 for t = 3 and A051240 for t = 4).
%F A000088 a(n, t) = Sum_{c : 1*c_1+2*c_2+...+n*c_n=n} per(c)*2^f(c), where:
%F A000088 ..per(c) = 1/(Product_{i=1..n} c_i!*i^c_i);
%F A000088 ..f(c) = (1/ord(c)) * Sum_{r=1..ord(c)} Sum_{x : 1*x_1+2*x_2+...+t*x_t=t} Product_{k=1..t} binomial(y(r, k; c), x_k);
%F A000088 ..ord(c) = lcm{i : c_i>0};
%F A000088 ..y(r, k; c) = Sum_{s|r : gcd(k, r/s)=1} s*c_(k*s) is the number of k-cycles of the r-th power of a permutation of type c. (End)
%F A000088 a(n) ~ 2^binomial(n,2)/n! [see Flajolet and Sedgewick p. 106, Gross and Yellen, p. 519, etc.]. - _N. J. A. Sloane_, Nov 11 2013
%F A000088 For asymptotics see also Lupanov 1959, 1960, also Turner and Kautz, p. 18. - _N. J. A. Sloane_, Apr 08 2014
%F A000088 a(n) = G(1) where G(z) = (1/n!) Sum_g det(I-g z^2)/det(I-g z) and g runs through the natural matrix n X n representation of the pair group A^2_n (for A^2_n see F. Harary and E. M. Palmer, Graphical Enumeration, page 83). - _Leonid Bedratyuk_, May 02 2015
%F A000088 From _Keith Briggs_, Jun 24 2016: (Start)
%F A000088 a(n) = 2^binomial(n,2)/n!*(
%F A000088 1+
%F A000088 2^( -n + 1)*n$2+
%F A000088 2^(-2*n + 3)*n$3*(n-7/3)+
%F A000088 2^(-3*n + 6)*n$4*(4*n^2/3 - 34*n/3 + 25) +
%F A000088 2^(-4*n + 10)*n$5*(8*n^3/3 - 142*n^2/3 + 2528*n/9 - 24914/45) +
%F A000088 2^(-5*n + 15)*n$6*(128*n^4/15 - 2296*n^3/9 + 25604*n^2/9 - 630554*n/45 + 25704) +
%F A000088 2^(-6*n + 21)*n$7*(2048*n^5/45 - 18416*n^4/9 + 329288*n^3/9 - 131680816*n^2/405 + 193822388*n/135 - 7143499196/2835) + ...),
%F A000088 where n$k is the falling factorial: n$k = n(n-1)(n-2)...(n-k+1), using the method of Wright 1969.
%F A000088 (End)
%F A000088 a(n) = 1/n*Sum_{k=1..n} a(n-k)*A003083(k). - _Andrey Zabolotskiy_, Aug 11 2020
%p A000088 # To produce all graphs on 4 nodes, for example:
%p A000088 with(GraphTheory):
%p A000088 L:=[NonIsomorphicGraphs](4,output=graphs,outputform=adjacency): # _N. J. A. Sloane_, Oct 07 2013
%p A000088 seq(GraphTheory[NonIsomorphicGraphs](n,output=count),n=1..10); # _Juergen Will_, Jan 02 2018
%p A000088 # alternative Maple program:
%p A000088 b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
%p A000088 +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
%p A000088 add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
%p A000088 end:
%p A000088 a:= n-> b(n$2, []):
%p A000088 seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 14 2019
%t A000088 Needs["Combinatorica`"]
%t A000088 Table[NumberOfGraphs[n], {n, 0, 19}] (* _Geoffrey Critzer_, Mar 12 2011 *)
%t A000088 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A000088 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
%t A000088 a[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t A000088 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 05 2018, after _Andrew Howroyd_ *)
%t A000088 b[n_, i_, l_] := If[n==0 || i==1, 1/n!*2^(Function[p, Sum[Ceiling[(p[[j]]-1 )/2]+Sum[GCD[p[[k]], p[[j]]], {k, 1, j-1}], {j, 1, Length[p]}]][Join[l, Table[1, {n}]]]), Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]];
%t A000088 a[n_] := b[n, n, {}];
%t A000088 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 03 2019, after _Alois P. Heinz_ *)
%o A000088 (Sage)
%o A000088 def a(n):
%o A000088 return len(list(graphs(n)))
%o A000088 # _Ralf Stephan_, May 30 2014
%o A000088 (PARI)
%o A000088 permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
%o A000088 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, v[i]\2)}
%o A000088 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 22 2017
%o A000088 (Python)
%o A000088 from itertools import combinations
%o A000088 from math import prod, factorial, gcd
%o A000088 from fractions import Fraction
%o A000088 from sympy.utilities.iterables import partitions
%o A000088 def A000088(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # _Chai Wah Wu_, Jul 02 2024
%Y A000088 Partial sums of A002494.
%Y A000088 Cf. A000666 (graphs with loops), A001349 (connected graphs), A002218, A006290, A003083.
%Y A000088 Column k=1 of A063841.
%Y A000088 Column k=2 of A309858.
%Y A000088 Row sums of A008406.
%Y A000088 Cf. also A000055, A000664.
%Y A000088 Partial sums are A006897.
%K A000088 core,nonn,nice
%O A000088 0,3
%A A000088 _N. J. A. Sloane_
%E A000088 Harary gives an incorrect value for a(8); compare A007149