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1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347
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OFFSET
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0,3
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COMMENTS
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The number of outcomes of chess tournament with n players.
For n >= 1 a(n) is the size of the Sylow 3-subgroup of the Chevalley group A_n(3) (sequence A053290). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
The number of binary relations on an n-element set that are both reflexive and antisymmetric. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The sequence a(n+1)= [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, .... - Philippe DELEHAM, Aug 29 2006
a(n) is the number of binary relations on a set with n elements that are total relations, i.e. for a relation on a set X it holds for all a and b in X that a~b or b~a (or both). E.g. a(2)= 3 because there are three total relations on a set with two elements: {(a,a),(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,b)}, {(a,a),(b,a),(b,b)}. - Geoffrey Critzer, May 23 2008
The number of semicomplete digraphs (or weak tournaments) on n labeled nodes. - Rémy-Robert Joseph, Nov 12 2012
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REFERENCES
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P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..65
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Index entries for sequences related to tournaments
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FORMULA
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a(n+1) is the determinant of n X n matrix M_(i, j)=binomial(3*i, j) - Benoit Cloitre, Aug 27 2003
Sequence is given by the Hankel transform (see A001906 for definition) of A007564 = {1, 1, 4, 19, 100, 562, 3304, ...}; example : det([1, 1, 4, 19; 1, 4, 19, 100; 4, 19, 100, 562; 19, 100, 562, 3304]) = 3^6 = 729 . - Philippe DELEHAM, Aug 20 2005
The sequence a(n+1)= [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, .., . - Philippe DELEHAM, Aug 29 2006
a(n)=3^(C(2+n,n)), n>=-2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007
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MAPLE
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seq(3^(binomial(2+n, n)), n=-2..12); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 16 2007
a:=n->mul (3^j, j=1..n): seq(a(n), n=-1..13); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
with(finance):seq(mul(futurevalue( 1, 2, k), k=0..n), n=-1..13); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008
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MATHEMATICA
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f[n_]:=3^n; lst={}; Do[a=f[n]; Do[a*=f[m], {m, n-1, 1, -1}]; AppendTo[lst, a], {n, 0, 20}]; lst [From Vladimir Joseph Stephan Orlovsky, Feb 10 2010]
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PROG
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(PARI) a(n)=3^binomial(n+1, 2) \\ Charles R Greathouse IV, Apr 17 2012
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CROSSREFS
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Cf. A007747.
Sequence in context: A085656 A113100 A038379 * A193610 A052269 A138525
Adjacent sequences: A047653 A047654 A047655 * A047657 A047658 A047659
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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