A094047 - OEIS

%I #55 Feb 16 2025 08:32:53

%S 0,0,2,12,312,9600,416880,23879520,1749363840,159591720960,

%T 17747520940800,2363738855385600,371511874881100800,

%U 68045361697964851200,14367543450324474009600,3464541314885011705344000,946263209467217020194816000,290616691739323132839591936000

%N Number of seating arrangements of n couples around a round table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together.

%C Also, the number of Hamiltonian directed circuits in the crown graph of order n.

%C Or the number of those 3 X n Latin rectangles (cf. A000186) the second row of which is a full cycle. - _Vladimir Shevelev_, Mar 22 2010

%D V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr.Mat.(J. of the Akademy of Sciences of Russia) 4(1992),91-110.

%H Seiichi Manyama, <a href="/A094047/b094047.txt">Table of n, a(n) for n = 1..253</a>

%H M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:<a href="http://doi.org/10.1007/978-3-319-44543-4_12">10.1007/978-3-319-44543-4_12</a>, arXiv:<a href="http://arxiv.org/abs/1510.07926">1510.07926</a>, [math.CO], 2015-2016.

%H H. M. Taylor, <a href="/A000179/a000179.pdf">A problem on arrangements</a>, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>

%F For n>1, a(n) = (-1)^n * 2 * (n-1)! + n! * Sum_{j=0..n-1} (-1)^j * (n-j-1)! * binomial(2*n-j-1,j). - _Max Alekseyev_, Feb 10 2008

%F a(n) = A059375(n) / (2*n) = A000179(n) * (n-1)!.

%F Conjecture: a(n) +(-n^2+2*n-3)*a(n-1) -(n-2)*(n^2-3*n+5)*a(n-2) -3*(n-2)*(n-3)*a(n-3) +(n-2)*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Nov 02 2015

%F Conjecture: (-n+2)*a(n) +(n-1)*(n^2-3*n+3)*a(n-1) +(n-2)*(n-1)*(n^2-3*n+3)*a(n-2) +(n-2)*(n-3)*(n-1)^2*a(n-3)=0. - _R. J. Mathar_, Nov 02 2015

%F a(n) = (n-1) * (n * (a(n-1) + a(n-2)) - 4 * (-1)^n * (n-3)!) for n > 3. - _Seiichi Manyama_, Jan 18 2020

%F a(n) = 2 * A306496(n). - _Alois P. Heinz_, Jun 19 2022

%p A094047 := proc(n)

%p     if n < 3 then

%p         0;

%p     else

%p         (-1)^n*2*(n-1)!+n!*add( (-1)^j*(n-j-1)!*binomial(2*n-j-1,j),j=0..n-1) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Nov 02 2015

%t Join[{0},Table[(-1)^n 2(n-1)!+n!Sum[(-1)^j (n-j-1)!Binomial[2n-j-1,j],{j,0,n-1}],{n,2,20}]] (* _Harvey P. Dale_, Mar 07 2012 *)

%Y Cf. A059375 (rotations are counted as different).

%Y Cf. A000179, A000186, A114939, A137729, A306496.

%K nonn

%O 1,3

%A _Matthijs Coster_, Apr 29 2004

%E Better definition from _Joel B. Lewis_, Jun 30 2007

%E Formula and further terms from _Max Alekseyev_, Feb 10 2008