OFFSET
1,3
COMMENTS
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
REFERENCES
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
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.)
FORMULA
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
MAPLE
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)
MATHEMATICA
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 *)
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) = {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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
a(1)-a(2) prepended by Andrew Howroyd, Sep 16 2018
STATUS
approved