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%I A000088 M1253 N0479
%S A000088 1,1,2,4,11,34,156,1044,12346,274668,12005168,1018997864,165091172592,
%T A000088 50502031367952,29054155657235488,31426485969804308768,64001015704527557894928,
%U A000088 245935864153532932683719776,1787577725145611700547878190848,24637809253125004524383007491432768
%N A000088 Number of 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000088 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000088 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 
               89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, 
               Annals of Discrete Math., 43 (1989), 89-102.
%D A000088 P. J. Cameron and C. R. Johnson, The number of equivalence patterns of 
               symmetric sign patterns, Discr. Math., 306 (2006), 3074-3077.
%D A000088 R. L. Davies, The numbers of structures of finite relations, Proc. Amer. 
               Math. Soc., 4 (1953), 486-494.
%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 S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
%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 W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. 
               Ann., 174 (1967), 53-78.
%D A000088 M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical 
               trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 
               563-567.
%D A000088 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A000088 R. W. Robinson, Enumeration of non-separable graphs, J. Combin. Theory 
               9 (1970), 327-356.
%D A000088 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A000088 A. Milicevic and N. Trinajstic, "Combinatorial Enumeration in Chemistry", 
               Chem. Modell., Vol. 4, (2006), pp. 405-469.
%H A000088 Keith M. Briggs, <a href="b000088.txt">Table of n, a(n) for n = 0..75</
               a> [From link below]
%H A000088 Keith M. Briggs, <a href="http://keithbriggs.info/cgt.html">Combinatorial 
               Graph Theory</a> [Gives first 140 terms]
%H A000088 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000088 E. Friedman, <a href="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 S. Hougardy, <a href="http://www2.informatik.hu-berlin.de/~hougardy/">
               Home Page</a>
%H A000088 Vladeta Jovovic, <a href="a063843.rtf">Formulae for the number T(n,k) 
               of n-multigraphs on k nodes</a>
%H A000088 Brendan McKay, <a href="http://www.csse.uwa.edu.au/~gordon/remote/graphs/
               graphcount">Maple program</a>.
%H A000088 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting 
               Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.2.
%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="a88.gif">Illustration of initial terms</a>
%H A000088 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SimpleGraph.html">Simple Graph</a>
%H A000088 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ConnectedGraph.html">Connected Graph</a>
%H A000088 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DegreeSequence.html">Degree Sequence</a>
%H A000088 Author not known, <a href="http://www.roard.com/docs/cookbook/cbsu5.html#x8-170001.5">
               Nonisomorphic graphs</a>.
%H A000088 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000088 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               105
%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 (vladeta(AT)eunet.rs) and Benoit Cloitre (benoit7848c(AT)orange.fr), 
               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 (keith.briggs(AT)bt.com), Oct 24 2005
%F A000088 Contribution from David Pasino (davepasino(AT)yahoo.com), Jan 31 2009: 
               (Start)
%F A000088 a(n) = a(n, 2) where a(n, t), the number of t-uniform hypergraphs on 
               n
%F A000088 unlabeled nodes (cf. A000665 for t = 3 and A051240 for t = 4), is
%F A000088 a(n, t) = (sum on c: 1*c_1+2*c_2+...+n*c_n= n) per(c)*2^f(c), where
%F A000088 per(c) = 1/(prod on i=1 to n) c_i!*i^c_i and f(c) = (1/ord(c)) *
%F A000088 (sum on r=1 to ord(c)) (sum on x: 1*x_1+2*x_2...+t*x_t=t) (prod on k 
               = 1 to t)
%F A000088 binom(y(r, k; c), x_k), where ord(c) = lcm{i : c_i > 0} and y(r, k; c) 
               =
%F A000088 (sum on s|r with gcd(k, r/s) = 1) s*c_(k*s) (= the number of k-cycles 
               of
%F A000088 the rth power of a permutation of type c). (End)
%Y A000088 Partial sums of A002494.
%Y A000088 Cf. A001349 (connected graphs), A002218, A006290. Second column of A063841. 
               Row sums of A008406.
%Y A000088 Sequence in context: A076320 A076321 A126149 this_sequence A071794 A107378 
               A086611
%Y A000088 Adjacent sequences: A000085 A000086 A000087 this_sequence A000089 A000090 
               A000091
%K A000088 core,nonn,nice
%O A000088 0,3
%A A000088 N. J. A. Sloane (njas(AT)research.att.com).
%E A000088 Harary gives an incorrect value for a(8). Compare A007149.

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