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 A047656 a(n) = 3^((n^2-n)/2). 26
 1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The number of outcomes of a 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 Deléham, 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)}, and {(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 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 . 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 REFERENCES P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375. LINKS 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. Joël Gay, Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018. T. R. Hoffman, J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014. G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2. FORMULA a(n+1) is the determinant of an n X n matrix M_(i, j) = C(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 Deléham, 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 Deléham, Aug 29 2006 a(n) = 3^binomial(n,2). - Zerinvary Lajos, Jun 16 2007 G.f. A(x) satisfies: A(x) = 1 + x * A(3*x). - Ilya Gutkovskiy, Jun 04 2020 EXAMPLE 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 MAPLE seq(3^binomial(n, 2), n=0..12); # Zerinvary Lajos, Jun 16 2007 a:= n-> mul(3^j, j=0..n-1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007 seq(3^((n^2-n)/2), n=0..14); MATHEMATICA 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 (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *) PROG (PARI) a(n)=3^binomial(n+1, 2) \\ Charles R Greathouse IV, Apr 17 2012 CROSSREFS Cf. A007747. Sequence in context: A085656 A113100 A038379 * A193610 A052269 A138525 Adjacent sequences:  A047653 A047654 A047655 * A047657 A047658 A047659 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)