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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.
5

%I #18 Jan 07 2020 17:30:46

%S 1,3,315,155925,212837625,618718975875,3287253918823875,

%T 28845653137679503125,388983632561608099640625,

%U 7637693625347175036443671875,209402646126143497974176151796875,7752714167528210725497923667975703125,377130780679409810741846496828678078515625

%N 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.

%C From _Karol A. Penson_, Oct 05 2009: (Start)

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

%C 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... .

%C This solution may not be unique. (End)

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

%H Andrew Howroyd, <a href="/A139541/b139541.txt">Table of n, a(n) for n = 0..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tournament.html">Tournament</a>

%H <a href="/index/To#tournament">Index entries for sequences related to tornaments</a>.

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

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

%F a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - _Reinhard Zumkeller_, Apr 28 2008

%o (PARI) a(n)={(4*n)!/(n!*8^n)} \\ _Andrew Howroyd_, Jan 07 2020

%Y Cf. A008299, A000142, A100733, A001018.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, Apr 25 2008

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 07 2020