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%I #27 Jun 14 2015 18:48:14
%S 1,6,1260,1247400,3405402000,19799007228000,210384250804728000,
%T 3692243601622976400000,99579809935771673508000000,
%U 3910499136177753618659160000000,214428309633170941925556379440000000
%N For n >= 1 a(n) is the size of the conjugacy class in the symmetric group S_(4n) consisting of permutations whose cycle decomposition is a product of n disjoint 4-cycles.
%C a(n) is the number of ways to seat 4n bridge players at n circular tables with four players at each table. - _Geoffrey Critzer_, Dec 17 2011
%H Harry J. Smith, <a href="/A060706/b060706.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = (4n)! / (n! * 4^n). Recursion: a(0) = 1, a(1) = 6, for n >= 2 a(n) = a(n-1) * C(4n - 1, 3)* 6 = a(n-1)*(4n-1)*(4n-2)*(4n-3). Using Stirling's formula in A000142 we have a(n) ~ 2 * 64^n * (n/e)^(3n).
%F E.g.f.: exp(x^4/4). - _Geoffrey Critzer_, Dec 17 2011
%F Write the generating function for this sequence in the form A(x) = sum_{n>=0} a(n)* x^(3*n+1)/(3*n+1)!. Then A'(x)*( 1 - A(x)^3) = 1, consequently A(x) is a root of z^4 - 4*z + 4*x with A(0) = 0. Cf. A052502. - _Peter Bala_, Jan 02 2015
%p for n from 0 to 20 do printf(`%d,`,(4*n)! / (n! * 4^n)) od:
%t nn = 40; a = x^4/4;f[list_] := Select[list, # > 0 &];
%t f[Range[0, nn]! CoefficientList[Series[Exp[a], {x, 0, nn}], x]] (* _Geoffrey Critzer_, Dec 17 2011 *)
%o (PARI) { for (n=0, 100, write("b060706.txt", n, " ", (4*n)! / (n! * 4^n)); ) } \\ _Harry J. Smith_, Jul 09 2009
%Y Cf. A000142. A001147, A052502, A052504.
%K nonn
%O 0,2
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
%E More terms from _James A. Sellers_, Apr 23 2001
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