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%I #40 Apr 23 2024 18:16:24
%S 1,2,6,60,960,24000,861840,42104160,2686763520,217039253760,
%T 21651071904000,2614084251609600,375698806311628800,
%U 63383303286471168000,12403896267489382656000,2786994829444848422400000,712575504763406361133056000
%N Number of circular permutations of 2n letters that are free of jealousy.
%C 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.
%C Limit_{n->oo} a(n)/(n-1)!^2 = Sum_{k>=1} 1/(k!*(k-1)!) = 1.590636854637329063382254424999666247954478159495536647132... (A096789).
%H Eiji Kurihara, <a href="http://kurihara.sansu.org/">Small room of mathematics</a>; see the answer for <a href="http://kurihara.sansu.org/sansu1-3/380.html">No. 380</a> of arithmetic challenges version 1.
%H Mikhail Kurkov and others, <a href="https://mathoverflow.net/q/469606">On a A089039 and pair of sequences with simple recursion</a>, Math Overflow, Apr 20 2024
%H Masaru Yoshikawa, <a href="http://www.sansu.org">Arithmetic challenges</a>. See problem No. 380.
%F 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.
%F a(n) = (n-1)!*(A001040(n-1) + A001053(n)) = 2*A276356(n), n > 1. - Conjectured by _Mikhail Kurkov_, Feb 10 2019 and proved by _Max Alekseyev_, Apr 23 2024 (see MO link)
%F a(n+4) = -(n+3)*(n+2)*(n*(n+1)*a(n) + 2*(n+1)^2*a(n+1) + n*(n+3)*a(n+2) - 2*a(n+3)) for all integer n>1. - conjectured by _Michael Somos_, Apr 21 2024. [The conjecture is equivalent to Kurkov's formula and thus is also proved. - _Max Alekseyev_, Apr 23 2024]
%e a(3)=6 because ABCcba, ACBbca, ABbacC, ACcabB, AabcCB, AacbBC are possible.
%t 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 *)
%o (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
%K nonn,nice,changed
%O 1,2
%A Akemi Nakamura, Michihiro Takahashi, Shogaku Meitantei (naka(AT)sansu.org), Dec 03 2003
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