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%I M1649 #47 Feb 27 2019 13:57:16
%S 1,1,2,6,20,86,662,8120,171526,5909259,348089533,33883250874,
%T 5476590066777,1490141905609371,666003784522738152,
%U 509204473666338077658,636051958071749028811326,1375164117171886868027357906,4844133410739656724629165903483,29777568550007746192195431057341474
%N Number of self-dual signed graphs with n nodes. Also number of self-complementary 2-multigraphs on n nodes.
%C A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).
%C Of a(1) through a(22) only a(3) = 2 is prime. - _Jonathan Vos Post_, Feb 19 2011
%D F. Harary and R. W. Robinson, Exposition of the enumeration of point-line-signed graphs, pp. 19 - 33 of Proc. Second Caribbean Conference Combinatorics and Computing (Bridgetown, 1977). Ed. R. C. Read and C. C. Cadogan. University of the West Indies, Cave Hill Campus, Barbados, 1977. vii+223 pp.
%D R. W. Robinson, personal communication.
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A004104/b004104.txt">Table of n, a(n) for n = 1..50</a> (terms 1..22 from R. W. Robinson)
%H Edward A. Bender and E. Rodney Canfield, <a href="https://doi.org/10.1016/0095-8956(83)90040-0">Enumeration of connected invariant graphs</a>, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 273.
%H Frank Harary, Edgar M. Palmer, Robert W. Robinson, Allen J. Schwenk, <a href="http://dx.doi.org/10.1002/jgt.3190010405">Enumeration of graphs with signed points and lines</a>, J. Graph Theory 1 (1977), no. 4, 295-308.
%H R. W. Robinson, <a href="/A000666/a000666_2.pdf">Notes - "A Present for Neil Sloane"</a>
%H R. W. Robinson, <a href="/A004102/a004102_1.pdf">Notes - computer printout</a>
%t permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t edges[v_] := Sum[Sum[If[Mod[v[[i]]*v[[j]], 2] == 0, GCD[v[[i]], v[[j]]], 0], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[If[Mod[v[[i]], 2] == 0, Quotient[v[[i]], 4]*2, 0], {i, 1, Length[v]}];
%t a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Feb 27 2019, after _Andrew Howroyd_ *)
%o (PARI)
%o permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, if(v[i]*v[j]%2==0, gcd(v[i],v[j])))) + sum(i=1, #v, if(v[i]%2==0, v[i]\4*2))}
%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ _Andrew Howroyd_, Sep 16 2018
%Y Cf. A004102, A052111, A052112, A052113.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jan 19 2000
%E a(18)-a(20) added by _Andrew Howroyd_, Sep 16 2018
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