1,3

Also, self-converse tournaments. - Brendan McKay, Dec 31 2020

Farrugia's Chapter 8 on enumeration of self-complementary and self-converse graphs and digraphs contains many explicit formulas as well as an in-depth discussion of the literature on this subject. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

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).

Andrew Howroyd, Table of n, a(n) for n = 1..100

Alastair Farrugia, Self-complementary graphs and generalizations: a comprehensive reference, M.Sc. Thesis, University of Malta, August 1999.

M. R. Sridharan, Self-complementary and self-converse oriented graphs , Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 1970 441-447. [Annotated scanned copy]

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 11 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

a(2*n) = Sum_{j partition of n & jk=0 if k even} [ Product_{k} 2^(k*jk^2-jk) * Product_{r<t} 2^(2*gcd(r, t)*jr*jt) / Product_{k} k^jk*jk! ]; a(2*n+1) = Sum_{j partition of n & jk=0 if k even} [ Product_{1<=r, t<=n} 2^(gcd(r, t)*jr*jt) / Product_{k} k^jk*jk! ]. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

with(combinat, partition): j:=proc(p) local k, jpart: jpart:=[seq(0, k=1..max(op(p)))]: for k from 1 to nops(p) do jpart[p[k]]:=jpart[p[k]]+1 od: RETURN(jpart): end; numeven:=jtot->2^add(add((2*igcd(r, t)*jtot[r]*jtot[t]), r=1..t-1)+(t*jtot[t]^2-jtot[t]), t=1..nops(jtot)); numodd:=jtot->mul(mul(2^(igcd(r, t)*jtot[r]*jtot[t]), r=1..nops(jtot)), t=1..nops(jtot)); den:=jtot->mul(k^jtot[k]*jtot[k]!, k=1..nops(jtot)); testj:=proc(jtot) local i: for i from 1 to floor(nops(jtot)/2) do if(jtot[2*i]<>0) then RETURN(0) fi od: RETURN(1) end; teven:=proc(n) local s, part, k, p, jtot: s:=0: part:=partition(n): for k from 1 to nops(part) do p:=part[k]: jtot:=j(p): if testj(jtot)=1 then s:=s+numeven(jtot)/den(jtot) fi od:RETURN(s): end; todd:=proc(n) local s, part, k, p, jtot: s:=0: part:=partition(n): for k from 1 to nops(part) do p:=part[k]: jtot:=j(p): if testj(jtot)=1 then s:=s+numodd(jtot)/den(jtot) fi od:RETURN(s): end; seq(op([todd(n), teven(n+1)]), n=1..12); (Pab Ter)

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_] := 2*Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i-1}], {i, 2, Length[v]}]+Total[v];

oddp[v_] := Module[{i}, For[i = 1, i <= Length[v], i++, If[BitAnd[v[[i]], 1] == 0, Return[0]]]; 1];

a[n_] := Module[{s = 0}, Do[If[oddp[p] == 1, s += permcount[2*p]*2^edges[p]*If[OddQ[n], n*2^Length[p], 1]], {p, IntegerPartitions[Quotient[n, 2]]}]; s/n!];

Array[a, 22] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

(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) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i])}

oddp(v) = {for(i=1, #v, if(bitand(v[i], 1)==0, return(0))); 1}

a(n) = {my(s=0); forpart(p=n\2, if(oddp(p), s+=permcount(2*Vec(p)) * 2^edges(p) * if(n%2, n*2^#p, 1))); s/n!} \\ Andrew Howroyd, Sep 16 2018

Cf. A000171, A003086, A005639.

Sequence in context: A089248 A006663 A094941 * A301603 A292038 A334600

Adjacent sequences: A002782 A002783 A002784 * A002786 A002787 A002788

nonn,nice,easy

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

a(1)-a(2) prepended by Andrew Howroyd, Sep 16 2018

approved