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%I M3063 N1241 #62 Apr 15 2021 20:34:34
%S 1,1,3,19,195,3031,67263,2086099,89224635,5254054111,426609529863,
%T 47982981969979,7507894696005795,1641072554263066471,
%U 502596525992239961103,216218525837808950623459,130887167385831881114006475,111653218763166828863141636911
%N Number of labeled connected bipartite graphs on n nodes.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%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 T. D. Noe, <a href="/A001832/b001832.txt">Table of n, a(n) for n = 1..50</a>
%H F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CJM-1979-007-3">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68.
%H F. Harary and R. W. Robinson, <a href="/A001832/a001832.pdf">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
%H D. A. Klarner, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80100-6">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19.
%H D. A. Klarner, <a href="/A000798/a000798_11.pdf">The number of graded partially ordered sets</a>, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
%H A. Nymeyer and R. W. Robinson, <a href="/A000684/a000684.pdf">Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs</a>, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/n-ColorableGraph.html">n-Colorable Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/n-ChromaticGraph.html">n-Chromatic Graph</a>
%F E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.
%F a(n) = A002031(n)/2, for n > 1. - _Geoffrey Critzer_, May 10 2011
%t mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* _Geoffrey Critzer_, May 10 2011 *)
%o (PARI) seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ _Andrew Howroyd_, Sep 26 2018
%Y Row sums of A228861.
%Y The unlabeled version is A005142.
%Y Cf. A002031, A047863, A047864.
%K nonn,nice,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 12 2003
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