

A047656


a(n) = 3^((n^2n)/2).


24



1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347
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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 3subgroup of the Chevalley group A_n(3) (sequence A053290).  Ahmed Fares (ahmedfares(AT)mydeja.com), Apr 30 2001
The number of binary relations on an nelement 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.  RémyRobert 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^2n)/2 nondiagonal 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^2n)/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, 344375.


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 TwoGraphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334, 2014
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Index entries for sequences related to tournaments


FORMULA

a(n+1) is the determinant of 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


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..n1): seq(a(n), n=0..12); # Zerinvary Lajos, Oct 03 2007
seq(3^((n^2n)/2), n=0..14);


MATHEMATICA

f[n_]:=3^n; lst={}; Do[a=f[n]; Do[a*=f[m], {m, n1, 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

N. J. A. Sloane


STATUS

approved



