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%I #109 Mar 10 2022 09:28:28
%S 1,1,3,27,729,59049,14348907,10460353203,22876792454961,
%T 150094635296999121,2954312706550833698643,
%U 174449211009120179071170507,30903154382632612361920641803529,16423203268260658146231467800709255289,26183890704263137277674192438430182020124347
%N a(n) = 3^((n^2-n)/2).
%C The number of outcomes of a chess tournament with n players.
%C 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
%C 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
%C 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 Deléham_, Aug 29 2006
%C 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)}, and {(a,a),(b,a),(b,b)}. - _Geoffrey Critzer_, May 23 2008
%C The number of semicomplete digraphs (or weak tournaments) on n labeled nodes. - _Rémy-Robert Joseph_, Nov 12 2012
%C The number of n X n binary matrices A that have a(i,j)=0 whenever a(j,i)=1 for i!=j and zeros on the diagonal. We need only consider the (n^2-n)/2 non-diagonal entry pairs <a(i,j), a(j,i)>. Since each pair is of the form <0,0>, <0,1>, or <1,0>, a(n) = 3^((n^2-n)/2). - _Dennis P. Walsh_, Apr 03 2014
%C a(n) is the number of symmetric (-1,0,1)-matrices of dimension (n-1) X (n-1). - _Eric W. Weisstein_, Jan 03 2021
%D P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.
%H Vincenzo Librandi, <a href="/A047656/b047656.txt">Table of n, a(n) for n = 0..65</a>
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H T. R. Hoffman and J. P. Solazzo, <a href="http://arxiv.org/abs/1408.0334">Complex Two-Graphs via Equiangular Tight Frames</a>, arXiv preprint arXiv:1408.0334 [math.CO], 2014.
%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/-101-Matrix.html">(-1,0,1)-Matrix</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SymmetricMatrix.html">Symmetric Matrix</a>
%H <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%F a(n+1) is the determinant of an n X n matrix M_(i, j) = C(3*i,j). - _Benoit Cloitre_, Aug 27 2003
%F 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 Deléham_, Aug 20 2005
%F 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 Deléham_, Aug 29 2006
%F a(n) = 3^binomial(n,2). - _Zerinvary Lajos_, Jun 16 2007
%F G.f. A(x) satisfies: A(x) = 1 + x * A(3*x). - _Ilya Gutkovskiy_, Jun 04 2020
%F a(n) = a(n-1)*3^(n-1), a(0) = 1. - _Mehdi Naima_, Mar 09 2022
%e The a(2)=3 binary 2 X 2 matrices are [0 0; 0 0], [0 1; 0 0], and [0 0; 1 0]. - _Dennis P. Walsh_, Apr 03 2014
%p seq(3^binomial(n, 2), n=0..12); # _Zerinvary Lajos_, Jun 16 2007
%p seq(3^((n^2-n)/2), n=0..14);
%t Table[3^((n^2 - n)/2), {n, 0, 14}] (* _Eric W. Weisstein_, Jan 03 2021 *)
%t 3^Table[Binomial[n, 2], {n, 0, 14}] (* _Eric W. Weisstein_, Jan 03 2021 *)
%t 3^Binomial[Range[0, 14], 2] (* _Eric W. Weisstein_, Jan 03 2021 *)
%t Table[Count[Tuples[{-1, 0, 1}, {n, n}], _?SymmetricMatrixQ], {n, 3}] (* _Eric W. Weisstein_, Jan 03 2021 *)
%o (PARI) a(n)=3^binomial(n+1,2) \\ _Charles R Greathouse IV_, Apr 17 2012
%Y Cf. A007747.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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