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 A000171 Number of self-complementary graphs with n nodes. (Formerly M0014 N0780) 16
 1, 0, 0, 1, 2, 0, 0, 10, 36, 0, 0, 720, 5600, 0, 0, 703760, 11220000, 0, 0, 9168331776, 293293716992, 0, 0, 1601371799340544, 102484848265030656, 0, 0, 3837878966366932639744, 491247277315343649710080, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) = A007869(n)-A054960(n), where A007869(n) is number of unlabeled graphs with n nodes and an even number of edges and A054960(n) is number of unlabeled graphs with n nodes and an odd number of edges. REFERENCES F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 139, Table 6.1.1. R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Andrew Howroyd, Table of n, a(n) for n = 1..100 H. Fripertinger, Self-complementary graphs Victoria Gatt, Mikhail Klin, Josef Lauri, Valery Liskovets, From Schur Rings to Constructive and Analytical Enumeration of Circulant Graphs with Prime-Cubed Number of Vertices, in Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, (Pilsen, Czechia, WAGT 2016) Vol. 305, Springer, Cham, 37-65. Richard A. Gibbs, Self-complementary graphs J. Combinatorial Theory Ser. B 16 (1974), 106--123. MR0347686 (50 #188). - N. J. A. Sloane, Mar 27 2012 Sebastian Jeon, Tanya Khovanova, 3-Symmetric Graphs, arXiv:2003.03870 [math.CO], 2020. B. D. McKay, Self-complementary graphs R. C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc., 38 (1963), 99-104. Eric Weisstein's World of Mathematics, Self-Complementary Graph D. Wille, Enumeration of self-complementary structures, J. Comb. Theory B 25 (1978) 143-150 FORMULA a(4n) = A003086(2n). a(4*n+1) = A047832(n), a(4*n+2) = a(4*n+3) = 0. - Andrew Howroyd, Sep 16 2018 MATHEMATICA < -1, {n, 1, 20}]  (* Geoffrey Critzer, Oct 21 2012 *) 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]; edges[v_] := 4 Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + 2 Total[v]; a[n_] := Module[{s = 0}, Switch[Mod[n, 4], 2|3, 0, _, Do[s += permcount[4 p]*2^edges[p]*If[OddQ[n], n*2^Length[p], 1], {p, IntegerPartitions[ Quotient[n, 4]]}]; s/n!]]; Array[a, 40] (* Jean-François Alcover, Aug 26 2019, after Andrew Howroyd *) PROG (PARI) 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} edges(v) = {4*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + 2*sum(i=1, #v, v[i])} a(n) = {my(s=0); if(n%4<2, forpart(p=n\4, s+=permcount(4*Vec(p)) * 2^edges(p) * if(n%2, n*2^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 16 2018 CROSSREFS Cf. A047660, A051251, A047832. Cf. A008406 (triangle of coefficients of the "graph polynomial"). Sequence in context: A181501 A213704 A278099 * A054922 A289651 A342588 Adjacent sequences:  A000168 A000169 A000170 * A000172 A000173 A000174 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from R. C. Read (rcread(AT)math.uwaterloo.ca) and Vladeta Jovovic STATUS approved

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Last modified April 20 01:36 EDT 2021. Contains 343117 sequences. (Running on oeis4.)