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A060706 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.

%I A060706 #32 Aug 05 2025 14:05:27
%S A060706 1,6,1260,1247400,3405402000,19799007228000,210384250804728000,
%T A060706 3692243601622976400000,99579809935771673508000000,
%U A060706 3910499136177753618659160000000,214428309633170941925556379440000000
%N A060706 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 A060706 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 A060706 Harry J. Smith, <a href="/A060706/b060706.txt">Table of n, a(n) for n = 0..100</a>
%H A060706 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 7.
%F A060706 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 A060706 E.g.f.: exp(x^4/4). - _Geoffrey Critzer_, Dec 17 2011
%F A060706 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 A060706 for n from 0 to 20 do printf(`%d,`,(4*n)! / (n! * 4^n)) od:
%t A060706 nn = 40; a = x^4/4;f[list_] := Select[list, # > 0 &];
%t A060706 f[Range[0, nn]! CoefficientList[Series[Exp[a], {x, 0, nn}], x]]  (* _Geoffrey Critzer_, Dec 17 2011 *)
%o A060706 (PARI) { for (n=0, 100, write("b060706.txt", n, " ", (4*n)! / (n! * 4^n)); ) } \\ _Harry J. Smith_, Jul 09 2009
%Y A060706 Cf. A000142. A001147, A052502, A052504.
%K A060706 nonn
%O A060706 0,2
%A A060706 Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
%E A060706 More terms from _James Sellers_, Apr 23 2001