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%I M1452 N0574 #45 Aug 15 2019 17:08:02
%S 0,0,1,2,5,13,44,191,1229,13588,288597,12297299,1031342116,
%T 166123498733,50668194387427,29104827043066808,31455591302381718651,
%U 64032471448906164191208,245999896712611657677614268,1787823725136869060356731751124
%N Number of disconnected graphs with n nodes.
%C a(n) is also the number of simple unlabeled graphs on n+1 nodes with diameter 2 and connectivity 1. - _Geoffrey Critzer_, Oct 23 2016
%D R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%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 Alois P. Heinz, <a href="/A000719/b000719.txt">Table of n, a(n) for n = 0..87</a> (first 31 terms from David Wasserman)
%H R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H F. Harary, <a href="http://dx.doi.org/10.1090/S0002-9947-1955-0068198-2">The number of linear, directed, rooted, and connected graphs</a>, Trans. Amer. Math. Soc. 78 (1955), 445-463.
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DisconnectedGraph.html">Disconnected Graph.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-ConnectedGraph.html">k-Connected Graph.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-Edge-ConnectedGraph.html">k-Edge-Connected Graph.</a>
%t << "Combinatorica`"; max = 18; A000088 = Table[ NumberOfGraphs[n], {n, 0, max}]; f[x_] = 1 - Product[ 1/(1 - x^k)^b[k], {k, 1, max}]; b[0] = b[1] = b[2] = 1; coes = CoefficientList[ Series[ f[x], {x, 0, max}], x]; sol = First[ Solve[ Thread[ Rest[ coes + A000088 ] == 0]]]; a[n_] := a[n] = A000088[[n+1]] - b[n] /. sol; a[0] = a[1] = 0; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Nov 24 2011 *)
%Y Equals (A000088) minus (A001349).
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_
%E Further terms from _Vladeta Jovovic_, Apr 14 2000
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