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 A000684 Number of colored labeled n-node graphs with 2 interchangeable colors. (Formerly M2954 N1192) 27
 1, 3, 13, 81, 721, 9153, 165313, 4244481, 154732801, 8005686273, 587435092993, 61116916981761, 9011561121239041, 1882834327457349633, 557257804202631217153, 233610656002563147038721, 138681207656726645785559041 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = A058872(n) + 1.  This sequence counts the empty graph on n nodes which is not allowed in A058872. - Geoffrey Critzer, Oct 07 2012 REFERENCES R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS R. W. Robinson and Vincenzo Librandi, Table of n, a(n) for n = 1..100 (first 32 terms from R. W. Robinson) S. R. Finch, Bipartite, k-colorable and k-colored graphs (2*A000684) S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author] F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy) D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy] A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.] R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976 FORMULA G.f.: A(x) = Sum_{n>=1} x^n/(1 - 2^n*x)^n. - Paul D. Hanna, Sep 14 2009 G.f.: 1/(W(0)-x) where W(k) = x*(x*2^k-1)^k - (x*2^(k+1)-1)^(k+1) + x*((2*x*2^k-1)^(2*k+2))/W(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 17 2012 From Peter Bala, Apr 01 2013: (Start) a(n) = Sum_{k = 0..n-1} binomial(n-1,k)*2^(k*(n-k)). a(n) = Sum_{k = 0..n} 2^k*A111636(n,k). Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence (but with an offset of 0) is E(x)*E(2*x) = Sum_{n >= 0} a(n+1)*x^n/(n!*2^C(n,2)) = 1 + 3*x + 13*x^2/(2!*2) + 81*x^3/(3!*2^3) + 721*x^4/(4!*2^6) + .... Cf. A134531. (End) MATHEMATICA With[{nn=20}, Rest[CoefficientList[Series[Sum[x^n/(1-2^n x)^n, {n, nn}], {x, 0, nn}], x]]] (* Harvey P. Dale, Nov 24 2011 *) PROG (PARI) a(n)=polcoeff(sum(k=1, n, x^k/(1-2^k*x +x*O(x^n))^k), n) \\ Paul D. Hanna, Sep 14 2009 CROSSREFS 2 * A000683(n) + 1. Cf. A058872, A111636, A134531. Sequence in context: A005923 A335588 A089461 * A222272 A057993 A208810 Adjacent sequences:  A000681 A000682 A000683 * A000685 A000686 A000687 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS a(15) onwards added by N. J. A. Sloane, Oct 19 2006 from the Robinson reference STATUS approved

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Last modified October 20 07:12 EDT 2021. Contains 348099 sequences. (Running on oeis4.)