OFFSET
0,3
COMMENTS
Number of permutations with n cycles on a set of size 2n.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
Asymptotic: a(n) ~ (2*n/(e*z*(1-z)))^n*sqrt((1-z)/(2*Pi*n*(2z-1))), where z=0.715331862959... is a root of the equation z = 2*(z-1)*log(1-z). - Vaclav Kotesovec, May 30 2011
Equivalent: a(n) ~ n!*(2*r^2/(r-1))^n/(2*Pi*n*sqrt(r-2)), where r=A226278. - Natalia L. Skirrow, Jul 13 2025
From Seiichi Manyama, May 20 2025: (Start)
a(n) = A132393(2*n,n).
a(n) = (2*n)! * [x^(2*n)] (-log(1 - x))^n / n!. (End)
MAPLE
seq(abs(Stirling1(2*n, n)), n=0..20);
MATHEMATICA
Table[Abs[StirlingS1[2n, n]], {n, 0, 12}]
N[1 + 1/(2 LambertW[-1, -Exp[-1/2]/2]), 50] (* The constant z in the asymptotic - Vladimir Reshetnikov, Oct 08 2016 *)
PROG
(Maxima) makelist(abs(stirling1(2*n, n)), n, 0, 12);
(PARI) for(n=0, 50, print1(abs(stirling(2*n, n, 1)), ", ")) \\ G. C. Greubel, Nov 09 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Mar 12 2011
STATUS
approved