A089039 Number of circular permutations of 2n letters that are free of jealousy.

1, 2, 6, 60, 960, 24000, 861840, 42104160, 2686763520, 217039253760, 21651071904000, 2614084251609600, 375698806311628800, 63383303286471168000, 12403896267489382656000, 2786994829444848422400000, 712575504763406361133056000

Offset

1,2

Comments

The number of circular permutations of 2*n people consisting of n married couples, such that no one sits next to a person of the opposite sex who is not his or her spouse.

Lim_{n->infinity} a(n)/(n-1)!^2 = Sum_{k>=1} 1/(k!*(k-1)!) = 1.590636854637329063382254424999666247954478159495536647132... (A096789).

Links

Table of n, a(n) for n=1..17.

Masaru Yoshikawa, Arithmetic challenges. See problem No. 380.

Eiji Kurihara, Small room of mathematics; see the answer for No. 380 of arithmetic challenges version 1.

Formula

a(1)=1, a(n) = Sum_{k=1..floor(n/2)} n!*(n-k-1)!^2/((k-1)!^2*(n-2*k)!*k) for n > 1.

a(n) = (n-1)!*(A001040(n-1) + A001053(n)) = 2*A276356(n), n > 1. - Mikhail Kurkov, Feb 10 2019

Example

a(3)=6 because ABCcba, ACBbca, ABbacC, ACcabB, AabcCB, AacbBC are possible.

Mathematica

a[1] = 1; a[n_] := n!*(n-2)!*HypergeometricPFQ[{1-n/2, 3/2-n/2}, {2, 2-n, 2-n}, 4]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Oct 30 2013, after symbolic sum *)

Prog

(PARI) a(n) = if (n==1, 1, sum (k=1, n\2, n!*(n-k-1)!^2/((k-1)!^2*(n-2*k)!*k))); \\ Michel Marcus, Sep 03 2013

Crossrefs

Sequence in context: A086332 A180402 A338600 * A156451 A152617 A156472

Adjacent sequences: A089036 A089037 A089038 * A089040 A089041 A089042

Keyword

nonn,nice

Author

Akemi Nakamura, Michihiro Takahashi, Shogaku Meitantei (naka(AT)sansu.org), Dec 03 2003

Status

approved