This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258665 #72 Sep 03 2023 23:40:01
%S A258665 0,0,0,1,5,20,116,791,6203,55000,543576,5922813,70518113,910704988,
%T A258665 12678282940,189251856883,3015212009143,51067548545968,
%U A258665 916175515710896,17355891466436025,346195661281979133,7252651426282955236,159210312386078554436,3654549974493252076175
%N A258665 A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular table, with no man seated next to his wife. After the ladies are seated at every other chair, M is the first man allowed to choose one of the remaining chairs. The sequence gives the number of ways of seating the other men, with no man seated next to his wife, if M chooses the chair that is 5 seats clockwise from his wife's chair.
%C A258665 This is a variation of the classic ménage problem (cf. A000179).
%C A258665 It is known [Riordan, ch. 8, ex. 7(b)] that, after the ladies are seated at every other chair, the number U_n of ways of seating the men in the ménage problem has asymptotic expansion U_n ~ e^(-2)*n!*(1 + Sum_{k>=1}(-1)^k/(k!(n-1)_k)), where (n)_k = n*(n-1)*...*(n-k+1).
%C A258665 Therefore, it is natural to conjecture that a(n) ~ e^(-2)*n!/(n-2)*(1 + Sum_{k>=1} (-1)^k/(k!(n-1)_k)).
%D A258665 I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124.
%D A258665 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, chs. 7, 8.
%H A258665 Peter J. C. Moses, <a href="/A258665/a258665.pdf">Seatings for 6 couples</a>.
%H A258665 I. Kaplansky and J. Riordan, <a href="/A000166/a000166_1.pdf">The problème des ménages</a>, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]
%H A258665 E. Lucas, <a href="https://archive.org/details/thoriedesnombre00lucagoog/page/n495">Sur le problème des ménages</a>, Théorie des nombres, Paris, 1891, 491-496.
%H A258665 Vladimir Shevelev, Peter J. C. Moses, <a href="http://arxiv.org/abs/1101.5321">The ménage problem with a known mathematician</a>, arXiv:1101.5321 [math.CO], 2011, 2015.
%H A258665 Vladimir Shevelev and Peter J. C. Moses, <a href="http://www.emis.de/journals/INTEGERS/papers/q72/q72.Abstract.html">Alice and Bob go to dinner: A variation on menage</a>, INTEGERS, Vol. 16(2016), #A72.
%H A258665 J. Touchard, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k31506/f631.image">Sur un problème de permutations</a>, C.R. Acad. Sci. Paris, 198 (1934), 631-633.
%F A258665 a(n) = Sum_{0<=k<=n-1} (-1)^k*(n-k-1)! * Sum_{max(k-n+3, 0)<=j<=min(k,2)} binomial(4-j, j)*binomial(2*n-k+j-6, k-j).
%t A258665 enumerateSeatings[pairs_,d_]:=If[d==1||d>=2pairs-1||EvenQ[d],{},
%t A258665 Map[#[[1]]&,DeleteCases[Map[{#,Differences[#]}&[Riffle[Range[pairs],#]]&,Map[Insert[#,1,(d+1)/2]&,Permutations[#,{Length[#]}]&[Rest[Range[pairs]]]]],{{___},{___,0,___}}]]];
%t A258665 enumerateSeatings[6,5]
%t A258665 a[pairs_,d_]:=If[pairs<=#-1||EvenQ[d]||d==1,0,Sum[((-1)^k)*(pairs-k-1)!Sum[Binomial[2#-j-4,j]*Binomial[2(pairs-#)-k+j+2,k-j],{j,Max[#+k-pairs-1,0],Min[k,#-2]}],{k,0,pairs-1}]]&[(d+3)/2];
%t A258665 Table[a[n,5],{n,15}] (* _Peter J. C. Moses_, Jun 13 2015 *)
%o A258665 (PARI) a(n) = sum(k=0,n-1, (-1)^k*(n-k-1)! * sum(j=max(k-n+3, 0), min(k,2), binomial(4-j, j)*binomial(2*n-k+j-6, k-j))); \\ _Michel Marcus_, Jun 13 2015
%Y A258665 Cf. A000179, A258664, A258666, A258667, A258673, A259212.
%K A258665 nonn
%O A258665 1,5
%A A258665 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 07 2015