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A002785
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Number of self-complementary oriented graphs with n nodes.
(Formerly M0375 N0141)
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1
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2, 2, 8, 12, 88, 176, 2752, 8784, 279968, 1492288, 95458560, 872687552, 111698291584, 1787154671104, 457509297625088, 13013584213369088, 6662951988432581120, 341143107490935724032, 349330527429800077778944
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,1
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COMMENTS
| 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
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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).
Sridharan, M. R.; Self-complementary and self-converse oriented graphs. Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 1970 441-447.
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LINKS
| Farrugia, Alastair; Self-complementary graphs and generalizations: a comprehensive reference, M.Sc. Thesis, University of Malta, August 1999.
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FORMULA
| a(2n) = sum_{j partition of n & jk=0 if k even} [ prod_{k} 2^(k*jk^2-jk) * prod_{r<t} 2^(2*gcd(r, t)*jr*jt) / prod_{k} k^jk*jk! ]; a(2n+1) = sum_{j partition of n & jk=0 if k even} [ prod_{1<=r, t<=n} 2^(gcd(r, t)*jr*jt) / prod_{k} k^jk*jk! ] - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
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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)
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CROSSREFS
| Sequence in context: A089248 A006663 A094941 * A045686 A045677 A005633
Adjacent sequences: A002782 A002783 A002784 * A002786 A002787 A002788
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
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