OFFSET
0,3
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, 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
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..100
G. Boccara, Nombre de représentations d'une permutation comme produit de deux cycles de longueurs données, Discrete Math., Vol. 29, No. 2 (1980), pp. 105-134.
Jean-Paul Doignon and Anthony Labarre, On Hultman Numbers, J. Integer Seq., Vol. 10 (2007), Article 07.6.2, 13 pages.
K. A. Penson and J.-M. Sixdeniers, Integral Representations of Catalan and Related Numbers, J. Integer Sequences, Vol. 4 (2001), Article 01.2.5.
Karol A. Penson and Allan I. Solomon, Coherent states from combinatorial sequences, arXiv:quant-ph/0111151, 2001.
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, Aug 27 2001
a(n) = n!^2*binomial(2*n,n)/(n+1). - Zerinvary Lajos, Jun 06 2006
a(n) = A090586(2*n + 1). - Gregory Gerard Wojnar, Jun 10 2021
From Amiram Eldar, Feb 08 2022: (Start)
Sum_{n>=0} 1/a(n) = cosh(1) + sinh(1)/2.
Sum_{n>=0} (-1)^n/a(n) = cos(1) - sin(1)/2. (End)
From Wolfdieter Lang, Feb 02 2024: (Start)
O.g.f.: hypergeometric([1,1,1,1/2],[2],4*x).
E.g.f.: hypergeometric([1,1,1/2],[2],4*x). (End)
a(n) = A177267(2n+1,n). - Alois P. Heinz, Feb 16 2024
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).
MAPLE
for n from 0 to 25 do printf(`%d, `, (2*n)!/(n+1)) od:
MATHEMATICA
Table[(2*n)!/(n + 1), {n, 0, 13}] (* Amiram Eldar, Feb 08 2022 *)
PROG
(PARI) { for (n=0, 100, write("b060593.txt", n, " ", (2*n)! / (n + 1)); ) } \\ Harry J. Smith, Jul 07 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
EXTENSIONS
More terms from James A. Sellers, Apr 13 2001
STATUS
approved