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A060593
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a(n) is the number of ways that a cycle of length 2n+1 in the symmetric group S_(2n+1) can be decomposed as the product of two cycles of length 2n+1.
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11
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1, 1, 8, 180, 8064, 604800, 68428800, 10897286400, 2324754432000, 640237370572800, 221172909834240000, 93666727314800640000, 47726800133326110720000, 28806532937614688256000000, 20325889640780924033433600000, 16578303738261941164769280000000
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OFFSET
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0,3
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COMMENTS
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The sequence deals only with S_m for odd m because for even m the number of representations of an m-cycle in S_m as a product of two m-cycles is zero.
a(n) = product of first 2n-1 numbers divided by their sum. E.g., a(3) = (1*2*3*4*5)/(1+2+3+4+5) = 120/15 = 8. - Amarnath Murthy, Jun 03 2004
a(n) is also the number of permutations in Sym(2n) whose "cycle graph" (or "breakpoint graph") contains exactly one alternating cycle, for n>=1 (see Doignon and Labarre). - Anthony Labarre, Jun 19 2007
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LINKS
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Jean-Paul Doignon and Anthony Labarre, On Hultman Numbers, J. Integer Seq., Vol. 10 (2007), Article 07.6.2, 13 pages.
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FORMULA
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a(n) = (2n)! / (n+1).
Integral representation as n-th moment of a positive function on positive half-axis, in Maple notation: a(n)=int(x^n*(exp(-sqrt(x))/sqrt(x)+Ei(-sqrt(x))), x=0..infinity), n=0, 1, 2, ..., where Ei(y) is the exponential integral. This representation is unique. - Karol A. Penson, Aug 27 2001
Sum_{n>=0} 1/a(n) = cosh(1) + sinh(1)/2.
Sum_{n>=0} (-1)^n/a(n) = cos(1) - sin(1)/2. (End)
O.g.f.: hypergeometric([1,1,1,1/2],[2],4*x).
E.g.f.: hypergeometric([1,1,1/2],[2],4*x). (End)
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EXAMPLE
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a(1) = 1 because in S_3 the only way to write the cycle (123) as a product of two 3-cycles is: (123) = (132)(132).
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MAPLE
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for n from 0 to 25 do printf(`%d, `, (2*n)!/(n+1)) od:
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MATHEMATICA
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Table[(2*n)!/(n + 1), {n, 0, 13}] (* Amiram Eldar, Feb 08 2022 *)
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PROG
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(PARI) { for (n=0, 100, write("b060593.txt", n, " ", (2*n)! / (n + 1)); ) } \\ Harry J. Smith, Jul 07 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
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EXTENSIONS
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STATUS
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approved
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