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%I #33 Jul 08 2016 17:40:41
%S 1,1,6,84,2160,89280,5443200,460857600,51819264000,7476605337600,
%T 1347105779712000,296508121620480000,78297264318873600000,
%U 24431729220414996480000,8893692297263669575680000,3735464765667589501747200000,1793050447716486548029440000000
%N Number of ways to arrange n women and n men around a circular table so that they can be divided into n nonintersecting pairs of 1 woman and 1 man sitting side-by-side.
%C a(n) is the number of monotonic paths from (0,0) to (n,n) which are equivalent to a path which is no more than 1 step from the main diagonal, where two paths are equivalent if they are circular shifts of one another.
%F a(n) = n!*(n-1)!*(2^n-1) for n>0, a(0) = 1.
%F a(n) = A000142(n)*A000142(n-1)*A000225(n), n >= 1. - _Omar E. Pol_, Jun 16 2016
%e For n = 2: Represent the women and men as the numbers 1,2,3,4, where 1,3 are women and 2,4 are men. Then all 6 arrangements around the circular table are valid (parentheses included to emphasize a valid pairing, including wraparound):
%e (1,2),(3,4);
%e (1,2),(4,3);
%e 1),(3,2),(4;
%e 1),(3,4),(2;
%e (1,4),(2,3);
%e (1,4),(3,2).
%o (PARI) a(n) = if (n==0, 1, n!*(n-1)!*(2^n-1)); \\ _Michel Marcus_, Jun 20 2016
%Y Cf. A000142, A000225, A010790.
%K nonn
%O 0,3
%A _Vincent Chan_, Jun 16 2016
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