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%I #39 Mar 02 2021 06:01:26
%S 1,1,2,6,18,78,522,6178,137352,6002584,509498932,82545586656,
%T 25251015686776,14527077828617744,15713242984902154384,
%U 32000507852263779299344,122967932076766466347469888,893788862572805850273939095424,12318904626562502262191503745716384
%N Number of complementary pairs of graphs on n nodes. Also number of unlabeled graphs with n nodes and an even number of edges.
%H Andrew Howroyd, <a href="/A007869/b007869.txt">Table of n, a(n) for n = 1..50</a>
%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 Nevena Francetić, Sarada Herke, and Ian M. Wanless, <a href="https://arxiv.org/abs/1703.04764">Parity of Sets of Mutually Orthogonal Latin Squares</a>, arXiv:1703.04764 [math.CO], 2017. See Section 4.1.
%H Tadeusz Sozański, <a href="https://doi.org/10.1002/jgt.3190040202">Enumeration of weak isomorphism classes of signed graphs</a>, J. Graph Theory 4 (1980), no. 2, 127-144. (<a href="https://www.zbmath.org/?q=ai%3Asozanski.tadeusz+se%3A00000494">Zentralblatt 434 #05059</a>)
%H Ferenc Szöllosi, <a href="https://arxiv.org/abs/1806.07861">The two-distance sets in dimension four</a>, arXiv:1806.07861 [math.MG], 2018. See Table 1.
%F Average of A000088 and A000171.
%t Needs["Combinatorica`"]; Table[Total[Table[NumberOfGraphs[n,m],{m,0,Binomial[n,2],2}]],{n,1,15}] (* _Geoffrey Critzer_, Oct 20 2012; modified by _Harvey P. Dale_, Aug 08 2013 *)
%o (PARI) a(n)={local(p=vector(n));
%o my(S=0, J() = sum(j=0, floor((n-1)/2), p[2*j+1]),
%o I2() = (sum(i=1, n, sum(j=1, n, p[i]*p[j]*gcd(i, j))) - J())/2,
%o M1() = (abs((p[1]-0)*(p[1]-1)) + sum(j=2, n, if(0!=(j%4), p[j], 0))),
%o inc()=!forstep(i=n, 1, -1, p[i]<n\i && p[i]++ && return; p[i]=0), t); until(inc(), t=0; for( i=1, n, if( n < t+=i*p[i], until(i++>n, p[i]=n); next(2))); t==n && S+=(if(M1() == 0, 2^I2()/prod(i=1, n, i^p[i]*p[i]!), 0) + 2^I2()/prod(i=1, n, i^p[i]*p[i]!))/2); S} \\ This is a modification of _M. F. Hasler_'s PARI program from A002854. - _Petros Hadjicostas_, Mar 02 2021
%Y Cf. A000088, A000171.
%Y Cf. A054960 for graphs with an odd number of edges.
%K nonn,nice
%O 1,3
%A _Peter J. Cameron_
%E More terms from _Vladeta Jovovic_, Jul 19 2000
%E Terms a(18) and beyond from _Andrew Howroyd_, Sep 17 2018
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