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

1,8

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).

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)

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 *)

Crossrefs

Cf. A177755, A177756, A177757, A177758, A177759, A178140, A187240

Sequence in context: A224587 A004670 A111044 * A204105 A015317 A195363

Adjacent sequences: A189786 A189787 A189788 * A189790 A189791 A189792

Keyword

nonn

Author

Vaclav Kotesovec, Apr 27 2011

Status

approved