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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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2
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1, 1, 8, 180, 8064, 604800, 68428800, 10897286400, 2324754432000, 640237370572800, 221172909834240000, 93666727314800640000, 47726800133326110720000, 28806532937614688256000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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 (amarnath_murthy(AT)yahoo.com), 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 (alabarre(AT)ulb.ac.be), Jun 19 2007
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REFERENCES
| G. Boccara, Nombre de representations d'une permutation comme produit de deux cycles de longueurs donnees, Discrete Math. 29 (1980), 105-134
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,100
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, 4 (2001), #01.2.5.
K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences.
J.-P. Doignon and A. Labarre, On Hultman Numbers, J. Integer Seq., 10 (2007), 13 pages.
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FORMULA
| 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 (penson(AT)lptl.jussieu.fr), Aug 27 2001
n!^2*(binomial(2*n,n)/(n+1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2006
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EXAMPLE
| 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
| for n from 0 to 25 do printf(`%d, `, (2*n)!/(n+1)) od:
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PROG
| (PARI) { for (n=0, 100, write("b060593.txt", n, " ", (2*n)! / (n + 1)); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 07 2009]
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CROSSREFS
| Sequence in context: A159326 A089456 A108552 * A130775 A203359 A180607
Adjacent sequences: A060590 A060591 A060592 * A060594 A060595 A060596
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KEYWORD
| nonn
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AUTHOR
| Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 13 2001
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