A189790 Number of ways to place n nonattacking bishops on an n X n toroidal board.

1, 4, 6, 64, 120, 2304, 5040, 147456, 362880, 14745600, 39916800, 2123366400, 6227020800, 416179814400, 1307674368000, 106542032486400, 355687428096000, 34519618525593600, 121645100408832000, 13807847410237440000

Offset: 1

Author

Vaclav Kotesovec, Apr 27 2011

Keywords

• nonn
• nice

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Crossrefs

Cf. A002465, A002454.

Programs

Mathematica

Table[If[EvenQ[n],2^n*((n/2)!)^2,n!],{n,1,20}]
Table[n!*SeriesCoefficient[1/(1-x)+x*ArcSin[x]/(1-x^2)^(3/2), {x,0,n}], {n,1,25}] (* Vaclav Kotesovec, Sep 26 2012 *)

Formula

a(n) = 2^n*((n/2)!)^2 if n is even and a(n) = n! if n is odd.

a(n) = n*(2*n-3)*a(n-2)-(n-3)*n*(n-2)^2*a(n-4). [Vaclav Kotesovec, Sep 26 2012]

E.g.f.: 1/(1-x)+x*arcsin(x)/(1-x^2)^(3/2). [Vaclav Kotesovec, Sep 26 2012]