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%I #15 Sep 08 2022 08:46:17
%S 2,3,6,20,80,420,2520,18144,145152,1330560,13305600,148262400,
%T 1779148800,23351328000,326918592000,4940103168000,79041650688000,
%U 1351612226764800,24329020081766400,464463110651904000,9289262213038080000,195848611658219520000
%N a(n) = n!/(m*(n-m)) with m = floor(n/2).
%C Consider this practical problem: n > 1 people are to be seated at two labeled round tables (T1 and T2), m of them at table T1, the rest at table T2. Two such seatings (A and B) are considered distinct if at least one person does not sit at the same table in seating A as in seating B, or has a different left or right neighbor (while rotating the seatings around any of the two tables is irrelevant). The number of such seatings is clearly binomial(n,m)*(m-1)!*(n-m-1)! which simplifies to this a(n). The formula holds for any m satisfying 0 < 2*m <= n.
%H Stanislav Sykora, <a href="/A277876/b277876.txt">Table of n, a(n) for n = 2..201</a>
%p a:= n-> (m->n!/(m*(n-m)))(floor(n/2)):
%p seq(a(n), n=2..30); # _Alois P. Heinz_, Nov 04 2016
%t Table[n! / (Floor[n/2] (n - Floor[n/2])), {n, 2, 25}] (* _Vincenzo Librandi_, Nov 04 2016 *)
%o (Magma) [Factorial(n)/(Floor(n/2)*(n-Floor(n/2))): n in [2..30]]; // _Vincenzo Librandi_, Nov 04 2016
%K nonn
%O 2,1
%A _Stanislav Sykora_, Nov 03 2016
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