A179427 Number of ways to place 7 nonattacking kings on an n X n toroidal board.
0, 0, 0, 0, 0, 3420, 576856, 19760512, 270487188, 2209065700, 12914201256, 59659859232, 231216019632, 781647658596, 2367858314700, 6553746728448, 16815788711212, 40446802230372, 92003239814224, 199311860224800, 413589922308360, 825997764087012, 1594007700404532, 2982430581363072, 5425904270482500, 9622254525739492, 16669554533555832, 28264133502586912, 46982453295836640, 76676963241363300
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
- Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).
Programs
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Mathematica
CoefficientList[Series[- 4 x^5 (1379 x^16 - 18219 x^15 + 124755 x^14 - 553765 x^13 + 1657983 x^12 - 3369984 x^11 + 4870575 x^10 - 6400905 x^9 + 10992208 x^8 - 19069951 x^7 + 21246441 x^6 - 8631071 x^5 - 7797385 x^4 + 8273322 x^3 + 2866693 x^2 + 131389 x + 855) / (x - 1)^15, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
Formula
Explicit formula: a(n) = 1/5040*n^2*(n^12 -189*n^10 +15295*n^8 -681135*n^6 +17692024*n^4 -255655596*n^2 +1617230880), n>=8.
G.f.: -4*x^6*(1379*x^16 - 18219*x^15 + 124755*x^14 - 553765*x^13 + 1657983*x^12 - 3369984*x^11 + 4870575*x^10 - 6400905*x^9 + 10992208*x^8 - 19069951*x^7 + 21246441*x^6 - 8631071*x^5 - 7797385*x^4 + 8273322*x^3 + 2866693*x^2 + 131389*x + 855)/(x-1)^15.