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A089039 Number of circular permutations of 2n letters that are free of jealousy. 1
1, 2, 6, 60, 960, 24000, 861840, 42104160, 2686763520, 217039253760, 21651071904000, 2614084251609600, 375698806311628800, 63383303286471168000, 12403896267489382656000, 2786994829444848422400000, 712575504763406361133056000 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 25 10:28 EDT 2021. Contains 345453 sequences. (Running on oeis4.)