login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A082368 a(n) = (4*n-1)! / (n! * n! * n! * (n-1)! * 3!). 4
1, 105, 15400, 2627625, 488864376, 96197645544, 19688264481600, 4148378852099625, 893864677761055000, 196056702961398759480, 43627992869961630486720, 9825387560922608865863400, 2235197406895366368301560000, 512889830640524227455318600000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of combinations that are possible when placing teams ranked #1 to #4*N in a single elimination tournament where there are four columns of N teams (as in the NCAA Men's Division-1 basketball tournament that is played in March) and each column is a separate regional tournament that produces one of the four semi-finalists. (The teams in the columns appear in sorted order and the relative positions of the four columns is irrelevant.)

Number of ways of dividing 4n labeled items into 4 unlabeled boxes with n items in each box. - Dan Parrish, Apr 09 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..300

FORMULA

a(n) = binomial(4*n,n)*binomial(3*n,n)*binomial(2*n,n)/24. - Zerinvary Lajos, Jun 25 2006

a(n) = (4n)!/(4!*n!^4). - Dan Parrish, Apr 09 2015

From Robert Israel, Apr 09 2015: (Start)

a(n) = Gamma(2*n+1/2)*Gamma(n+1/2)*64^n/(24*Pi*(n!)^3).

a(n+1) = 8*(2*n+1)*(4*n+1)*(4*n+3)*a(n)/(n+1)^3.

G.f.: g(x) = x*hypergeom([1,5/4,3/2,7/4],[2,2,2],256*x) satisfies

x^4*(256*x-1)*g''''(x) + 5*x^3*(384*x-1)*g'''(x) + 4*x^2*(780*x-1)*g''(x) + 840*x^2*g'(x) = 0. (End)

EXAMPLE

8 ranked teams (n=2) in a four region, single elimination tournament generates 105 different possible tournament orderings, where the teams in each region are ordered from best to worst. (Teams would be matched up from top to bottom and continue towards the middle two for other matchups, when more than two teams are listed in each column.) 105 tournaments is too many to list here. As this formula applies to single elimination tournaments, this enumeration formula really only makes sense when n is even.

MAPLE

[seq(binomial(4*n, n)*binomial(3*n, n)*binomial(2*n, n)/24, n=1..17)]; # Zerinvary Lajos, Jun 25 2006

MATHEMATICA

Table[(4 n)! / (4! n!^4), {n, 30}] (* Vincenzo Librandi, Jun 16 2017 *)

PROG

(PARI) a(n)=(4*n)!/(4!*n!^4) \\ Charles R Greathouse IV, Apr 09 2015

(MAGMA) [Factorial(4*n-1) / (Factorial(n)*Factorial(n)* Factorial(n)*Factorial(n-1)*6): n in [1..15]]; // Vincenzo Librandi, Jun 16 2017

CROSSREFS

Row 4 of A060540.

Cf. A000984, A005809, A005810.

Sequence in context: A255497 A094075 A199519 * A001546 A111647 A295463

Adjacent sequences:  A082365 A082366 A082367 * A082369 A082370 A082371

KEYWORD

easy,nonn

AUTHOR

John A. Trono (jtrono(AT)smcvt.edu), May 10 2003

EXTENSIONS

More terms from Zerinvary Lajos, Jun 25 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 22 16:57 EDT 2022. Contains 353957 sequences. (Running on oeis4.)