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Number of ways to place n nonattacking bishops on an 2n x 2n toroidal board.
3

%I #13 Sep 12 2015 11:00:25

%S 4,80,2688,132864,8647680,699678720,67711795200,7629571031040,

%T 981168437329920,141817953779712000,22760391875493888000,

%U 4016046336733347840000,772743693378451931136000,161027573368536472485888000,36127883615009765477842944000

%N Number of ways to place n nonattacking bishops on an 2n x 2n toroidal board.

%H Vincenzo Librandi, <a href="/A189791/b189791.txt">Table of n, a(n) for n = 1..200</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens, kings, bishops and knights</a> (in English and Czech)

%F a(n)=2^n*n!*Sum[Binomial[n,i]^3,{i,0,n}].

%F Asymptotic: a(n) ~ 2^(4n+1)*(n-1)!/Pi/sqrt(3) ~ 2^(4n+1)*n^n/exp(n)*sqrt(2/(3*Pi*n)).

%F Recurrence: a(n) = ((14*n^2-14*n+4)*a(n-1) + 32*(n-1)^3*a(n-2))/n.

%t Table[2^n*n!*Sum[Binomial[n,i]^3,{i,0,n}],{n,1,20}]

%Y Cf. A189790, A002465, A000172.

%K nonn

%O 1,1

%A _Vaclav Kotesovec_, Apr 27 2011