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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.

%I A139541 #19 Feb 16 2025 08:33:08
%S A139541 1,3,315,155925,212837625,618718975875,3287253918823875,
%T A139541 28845653137679503125,388983632561608099640625,
%U A139541 7637693625347175036443671875,209402646126143497974176151796875,7752714167528210725497923667975703125,377130780679409810741846496828678078515625
%N 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.
%C A139541 From _Karol A. Penson_, Oct 05 2009: (Start)
%C A139541 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 A139541 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 A139541 This solution may not be unique. (End)
%D A139541 G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Appendix: Problem 203.1, p164.
%H A139541 Andrew Howroyd, <a href="/A139541/b139541.txt">Table of n, a(n) for n = 0..100</a>
%H A139541 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Tournament.html">Tournament</a>
%H A139541 <a href="/index/To#tournament">Index entries for sequences related to tornaments</a>.
%F A139541 a(n) = (4*n)!/(n!*8^n).
%F A139541 a(n) = A001147(n)*A001147(2*n).
%F A139541 a(n) = A008977(n)*(A049606(n)/A001316(n))^3. - _Reinhard Zumkeller_, Apr 28 2008
%o A139541 (PARI) a(n)={(4*n)!/(n!*8^n)} \\ _Andrew Howroyd_, Jan 07 2020
%Y A139541 Cf. A008299, A000142, A100733, A001018.
%K A139541 nonn
%O A139541 0,2
%A A139541 _Reinhard Zumkeller_, Apr 25 2008
%E A139541 Terms a(11) and beyond from _Andrew Howroyd_, Jan 07 2020