0,3

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

G. Boccara, Nombre de representations d'une permutation comme produit de deux cycles de longueurs donnees, Discrete Math. 29 (1980), 105-134

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.

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

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)

for n from 0 to 25 do printf(`%d, `, (2*n)!/(n+1)) od:

Sequence in context: A001596 A089456 A108552 this_sequence A130775 A024286 A049034

Adjacent sequences: A060590 A060591 A060592 this_sequence A060594 A060595 A060596

nonn

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

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 13 2001