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%I M2486 N0985 #65 Nov 14 2022 20:03:04
%S 1,1,1,3,5,12,30,79,227,710,2322,8071,29503,112822,450141,1867871,
%T 8037472,35787667,164551477,779945969,3804967442,19079312775,
%U 98211456209,518397621443,2802993986619,15510781288250,87765472487659,507395402140211,2994893000122118,18035546081743772,110741792670074054,692894304050453139
%N Number of connected graphs with n edges.
%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 Max Alekseyev, <a href="/A002905/b002905.txt">Table of n, a(n) for n = 0..60</a>
%H G. A. Baker et al., <a href="http://dx.doi.org/10.1103/PhysRev.164.800">High-temperature expansions for the spin-1/2 Heisenberg model</a>, Phys. Rev., 164 (1967), 800-817.
%H Nicolas Borie, <a href="http://arxiv.org/abs/1511.05843">The Hopf Algebra of graph invariants</a>, arXiv preprint arXiv:1511.05843 [math.CO], 2015.
%H P. J. Cameron, <a href="https://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 Mike Cummings and Adam Van Tuyl, <a href="https://arxiv.org/abs/2211.02471">The GeometricDecomposability package for Macaulay2</a>, arXiv:2211.02471 [math.AC], 2022.
%H Anjan Dutta and Hichem Sahbi, <a href="https://arxiv.org/abs/1803.00425">Graph Kernels based on High Order Graphlet Parsing and Hashing</a>, arXiv:1803.00425 [cs.CV], 2018.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/graphs/">Small graphs</a>
%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 Peter Steinbach, <a href="/A000664/a000664_1.pdf">Field Guide to Simple Graphs, Volume 4</a>, Part 1 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polynema.html">Polynema.</a>
%F A000664 and this sequence are an Euler transform pair. - _N. J. A. Sloane_, Aug 30 2016
%e a(3) = 3 since the three connected graphs with three edges are a path, a triangle and a "Y".
%e The first difference between this sequence and A046091 is for n=9 edges where we see K_{3,3}, the well-known "utility graph".
%t A000664 = Cases[Import["https://oeis.org/A000664/b000664.txt", "Table"], {_, _}][[All, 2]];
%t (* EulerInvTransform is defined in A022562 *)
%t Join[{1}, EulerInvTransform[Rest @ A000664]] (* _Jean-François Alcover_, May 10 2019, updated Mar 17 2020 *)
%Y Column sums of A054924 or equivalently row sums of A054923.
%Y Cf. A000664, A046091 (for connected planar graphs), A275421 (multisets).
%Y Apart from a(3), same as A003089.
%K nonn,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jan 12 2000
%E More terms from _Gordon F. Royle_, Jun 05 2003
%E a(25)-a(26) from _Max Alekseyev_, Sep 19 2009
%E a(27)-a(60) from _Max Alekseyev_, Sep 07 2016
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