A215943 Number of ways to place k non-attacking bishops on an n x n toroidal chessboard, summed over all k >= 0.

2, 9, 34, 289, 1546, 19321, 130922, 2169729, 17572114, 364466281, 3405357682, 85143154849, 896324308634, 26309790300249, 306827170866106, 10366719612433921, 132240988644215842, 5064730099043043529, 69974827707903049154, 3000912883089564050721

Offset: 1

Author

Vaclav Kotesovec, Aug 28 2012

Keywords

• nonn

Comments

a(n) = A002720(n) if n is odd.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..440
• V. Kotesovec, Non-attacking chess pieces

Crossrefs

Cf. A002720, A201862, A010790, A177755-A177759, A178140, A189789, A189790, A189791.

Programs

Mathematica

Table[Sum[If[EvenQ[n],2^k*k!*Sum[Binomial[n/2,i]^2*Binomial[n/2,k-i]^2/Binomial[k,i],{i,0,k}],Binomial[n,k]^2*k!],{k,0,n}],{n,1,25}]

Formula

Recurrence: a(n) = ((12*n^5 - 158*n^4 - (6*(-1)^n-706)*n^3 - (1193-41*(-1)^n)*n^2 - 8*(7*(-1)^n-72)*n - 22*(-1)^n-28)*a(n-2) + (-12*n^6 + 206*n^5 + 2*(7*(-1)^n-691)*n^4 + (4545-137*(-1)^n)*n^3 + (442*(-1)^n-7442)*n^2 + (5194-544*(-1)^n)*n + 198*(-1)^n-698)*(n-2)*a(n-4) + 2*(2*n-1)*(n^2-7*n+10)^2*(n-4)^4*a(n-6))/(2*(n-5)^2*(2*n-5)).