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 A094047 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. 19
 0, 0, 2, 12, 312, 9600, 416880, 23879520, 1749363840, 159591720960, 17747520940800, 2363738855385600, 371511874881100800, 68045361697964851200, 14367543450324474009600, 3464541314885011705344000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also, the number of Hamiltonian directed circuits in the crown graph of order n. 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 REFERENCES 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. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..253 M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:10.1007/978-3-319-44543-4_12,  arXiv:1510.07926, [math.CO], 2015-2016. H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy] Eric Weisstein's World of Mathematics, Crown Graph Eric Weisstein's World of Mathematics, Hamiltonian Cycle FORMULA 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 a(n) = A059375(n) / (2*n) = A000179(n) * (n-1)!. 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 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 a(n) = (n-1) * (n * (a(n-1) + a(n-2)) - 4 * (-1)^n * (n-3)!) for n > 3. - Seiichi Manyama, Jan 18 2020 MAPLE A094047 := proc(n)     if n < 3 then         0;     else         (-1)^n*2*(n-1)!+n!*add( (-1)^j*(n-j-1)!*binomial(2*n-j-1, j), j=0..n-1) ;     end if; end proc: # R. J. Mathar, Nov 02 2015 MATHEMATICA 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 *) CROSSREFS Cf. A059375 (rotations are counted as different) Cf. A000179, A114939, A137729. Sequence in context: A012422 A122767 A260321 * A300045 A091472 A156518 Adjacent sequences:  A094044 A094045 A094046 * A094048 A094049 A094050 KEYWORD nonn AUTHOR Matthijs Coster, Apr 29 2004 EXTENSIONS Better definition from Joel B. Lewis, Jun 30 2007 Formula and further terms from Max Alekseyev, Feb 10 2008 STATUS approved

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Last modified May 13 00:11 EDT 2021. Contains 343829 sequences. (Running on oeis4.)