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A060706
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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.
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7
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1, 6, 1260, 1247400, 3405402000, 19799007228000, 210384250804728000, 3692243601622976400000, 99579809935771673508000000, 3910499136177753618659160000000, 214428309633170941925556379440000000
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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
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MAPLE
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for n from 0 to 20 do printf(`%d, `, (4*n)! / (n! * 4^n)) od:
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MATHEMATICA
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nn = 40; a = x^4/4; f[list_] := Select[list, # > 0 &];
f[Range[0, nn]! CoefficientList[Series[Exp[a], {x, 0, nn}], x]] (* Geoffrey Critzer, Dec 17 2011 *)
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PROG
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(PARI) { for (n=0, 100, write("b060706.txt", n, " ", (4*n)! / (n! * 4^n)); ) } \\ Harry J. Smith, Jul 09 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
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EXTENSIONS
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STATUS
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approved
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