

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.


13



0, 0, 2, 12, 312, 9600, 416880, 23879520, 1749363840, 159591720960, 17747520940800, 2363738855385600, 371511874881100800, 68045361697964851200, 14367543450324474009600, 3464541314885011705344000
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OFFSET

1,3


COMMENTS

Also, the number of Hamiltonian directed circuits in the crown graph of order n.
Or the number of those 3Xn Latin rectangles (cf. A000186) the second row of which is a full cycle. [From 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),91110. [From Vladimir Shevelev, Mar 22 2010]


LINKS

Table of n, a(n) for n=1..16.
Max Alekseyev, On enumeration of seating arrangements of couples around a circular table, Automatic Sequences Workshop, 2015
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 * (n1)! + n! * SUM[j=0..n1] (1)^j * (nj1)! * binomial(2*nj1,j).  Max Alekseyev, Feb 10 2008
a(n) = A059375(n) / (2*n) = A000179(n) * (n1)!.
Conjecture: a(n) +(n^2+2*n3)*a(n1) (n2)*(n^23*n+5)*a(n2) 3*(n2)*(n3)*a(n3) +(n2)*(n3)*(n4)*a(n4)=0.  R. J. Mathar, Nov 02 2015
Conjecture: (n+2)*a(n) +(n1)*(n^23*n+3)*a(n1) +(n2)*(n1)*(n^23*n+3)*a(n2) +(n2)*(n3)*(n1)^2*a(n3)=0.  R. J. Mathar, Nov 02 2015


MAPLE

A094047 := proc(n)
if n < 3 then
0;
else
(1)^n*2*(n1)!+n!*add( (1)^j*(nj1)!*binomial(2*nj1, j), j=0..n1) ;
end if;
end proc: # R. J. Mathar, Nov 02 2015


MATHEMATICA

Join[{0}, Table[(1)^n 2(n1)!+n!Sum[(1)^j (nj1)!Binomial[2nj1, j], {j, 0, n1}], {n, 2, 20}]] (* Harvey P. Dale, Mar 07 2012 *)


CROSSREFS

Cf. A059375 (rotations are counted as different)
Cf. A114939, A137729.
Sequence in context: A012422 A122767 A260321 * A091472 A156518 A012727
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



