A139541 There are 4*n players who wish to play bridge at n tables. Each player must have another player as partner and each pair of partners must have another pair as opponents. The choice of partners and opponents can be made in exactly a(n)=(4*n)!/(n!*8^n) different ways.

1, 3, 315, 155925, 212837625, 618718975875, 3287253918823875, 28845653137679503125, 388983632561608099640625, 7637693625347175036443671875, 209402646126143497974176151796875, 7752714167528210725497923667975703125, 377130780679409810741846496828678078515625

Offset: 0

Author

Reinhard Zumkeller, Apr 25 2008

Keywords

• nonn

Comments

From Karol A. Penson, Oct 05 2009: (Start)

Integral representation as n-th moment of a positive function on a positive semi-axis (solution of the Stieltjes moment problem), in Maple notation:

a(n)=int(x^n*((1/4)*sqrt(2)*(Pi^(3/2)*2^(1/4)*hypergeom([], [1/2, 3/4], -(1/32)*x)*sqrt(x)-2*Pi*hypergeom([], [3/4, 5/4], -(1/32)*x)*GAMMA(3/4)*x^(3/4)+sqrt(Pi)*GAMMA(3/4)^2*2^(1/4)*hypergeom([], [5/4, 3/2],-(1/32)*x)*x)/(Pi^(3/2)*GAMMA(3/4)*x^(5/4))), x=0..infinity), n=0,1... .

This solution may not be unique. (End)

References

• G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Appendix: Problem 203.1, p164.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, Tournament
• Index entries for sequences related to tornaments.

Crossrefs

Cf. A008299, A000142, A100733, A001018.

Programs

PARI

a(n)={(4*n)!/(n!*8^n)} \\ Andrew Howroyd, Jan 07 2020

Formula

a(n) = (4*n)!/(n!*8^n).

a(n) = A001147(n)*A001147(2*n).

a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - Reinhard Zumkeller, Apr 28 2008

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 07 2020