A060593 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.

1, 1, 8, 180, 8064, 604800, 68428800, 10897286400, 2324754432000, 640237370572800, 221172909834240000, 93666727314800640000, 47726800133326110720000, 28806532937614688256000000, 20325889640780924033433600000, 16578303738261941164769280000000

Offset: 0

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001

Keywords

• nonn
• easy

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

Examples

			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).
		

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.
• Ivan V. Morozov, On Quotients of a More General Theorem of Wilson, arXiv:2505.16201 [math.NT], 2025. See Z p. 9. Z(2*n+1)=1+a(n), Z(2n)=1.
• Karol 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.

Crossrefs

Cf. A090586, A177267.

Programs

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 *)

PARI

a(n) = (2*n)! / (n + 1); \\ Harry J. Smith, Jul 07 2009

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

D-finite with recurrence (n+1)*a(n) -2*n^2*(2*n-1)*a(n-1)=0. - R. J. Mathar, May 26 2025

Extensions

More terms from James Sellers, Apr 13 2001