A189789 Number of ways to place 8 nonattacking bishops on an n x n toroidal board.
0, 0, 0, 0, 0, 0, 0, 147456, 3265920, 129024000, 1097712000, 12939264000, 66784798080, 436483031040, 1669619952000, 7629571031040, 23828156352000, 85476013572096, 230333593351680, 693478195200000, 1669577821632000
Offset: 1
Keywords
Links
- Bruno Berselli, Table of n, a(n) for n = 1..5000
- V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
- Index entries for linear recurrences with constant coefficients, signature (2, 14, -30, -90, 210, 350, -910, -910, 2730, 1638, -6006, -2002, 10010, 1430, -12870, 0, 12870, -1430, -10010, 2002, 6006, -1638, -2730, 910, 910, -350, -210, 90, 30, -14, -2, 1).
Programs
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Mathematica
(* Number of ways to place k nonattacking bishops on an n x n toroidal board *) tbishops[k_,n_]:=If[EvenQ[n],2^k*k!*Sum[Binomial[n/2,i]^2*Binomial[n/2,k-i]^2/Binomial[k,i],{i,0,k}],k!*Binomial[n,k]^2]; Table[tbishops[8,n],{n,1,20}] (* using k=8 for this sequence *)
Formula
a(n) = (1/80640) * (n-6)^2 * (n-4)^2 * (n-2)^2 * n^2 * (2*n^8 - 64*n^7 + 884*n^6 - 7048*n^5 + 37382*n^4 - 147904*n^3 + 468540*n^2 - 1108800*n + 1422225 + (28*n^6 - 840*n^5 + 10906*n^4 - 80640*n^3 + 370468*n^2 - 1034880*n + 1400175) * (-1)^n)
G.f.: 1152x^8*(35x^23 + 21178x^22 + 27889x^21 + 2133348x^20 + 3081175x^19 + 51948910x^18 + 72476645x^17 + 469213640x^16 + 538879520x^15 + 1803221880x^14 + 1580004720x^13 + 3146148264x^12 + 2014875632x^11 + 2544618104x^10 + 1144092320x^9 + 933224520x^8 + 278242005x^7 + 143723790x^6 + 25756935x^5 + 7854820x^4 + 693025x^3 + 104538x^2 + 2579x + 128) / ((1-x)^17*(x+1)^15)