A273935 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.

1, 1, 6, 84, 2160, 89280, 5443200, 460857600, 51819264000, 7476605337600, 1347105779712000, 296508121620480000, 78297264318873600000, 24431729220414996480000, 8893692297263669575680000, 3735464765667589501747200000, 1793050447716486548029440000000

Offset: 0

Author

Vincent Chan, Jun 16 2016

Keywords

• nonn

Comments

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.

Examples

			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):
(1,2),(3,4);
(1,2),(4,3);
1),(3,2),(4;
1),(3,4),(2;
(1,4),(2,3);
(1,4),(3,2).
		

Crossrefs

Cf. A000142, A000225, A010790.

Programs

PARI

a(n) = if (n==0, 1, n!*(n-1)!*(2^n-1)); \\ Michel Marcus, Jun 20 2016

Formula

a(n) = n!*(n-1)!*(2^n-1) for n>0, a(0) = 1.

a(n) = A000142(n)*A000142(n-1)*A000225(n), n >= 1. - Omar E. Pol, Jun 16 2016