A173436 - OEIS

%I #11 Sep 12 2015 11:00:22

%S 0,0,0,16,0,80352,1359288,31404480,339256836,2527519400,14053530964,

%T 63100177488,240356217660,803630856504,2416671974700,6655251717376,

%U 17015566051020,40822003107000,92679987456312,200490192134800

%N Number of ways to place 7 nonattacking knights on an n X n toroidal board.

%H Vincenzo Librandi, <a href="/A173436/b173436.txt">Table of n, a(n) for n = 1..1000</a>

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>

%F Explicit formula: a(n) = n^2*(n^12-189n^10+16135n^8-801255n^6+24595984n^4-445931556n^2+3756080880)/5040, n>=14. For any fixed value of k > 1, a(n) = n^(2k)/k! - 9n^(2k-2)/2/(k-2)! + (243k+143)*n^(2k-4)/24/(k-3)! - ...

%F G.f.: -4*x^4 * (2535*x^24 -61497*x^23 +627330*x^22 -3849410*x^21 +16791330*x^20 -58053150*x^19 +170691269*x^18 -438580125*x^17 +976505385*x^16 -1844050487*x^15 +2900976825*x^14 -3760563305*x^13 +3991133690*x^12 -3450574470*x^11 +2418714751*x^10 -1370750375*x^9 +628081926*x^8 -228075638*x^7 +56855445*x^6 -6423333*x^5 +4868490*x^4 +36682*x^3 +20508*x^2 -60*x +4) / (x-1)^15. [_Vaclav Kotesovec_, Mar 25 2010]

%t CoefficientList[Series[- 4 x^3 (2535 x^24 - 61497 x^23 + 627330 x^22 - 3849410 x^21 + 16791330 x^20 - 58053150 x^19 + 170691269 x^18 - 438580125 x^17 + 976505385 x^16 - 1844050487 x^15 + 2900976825 x^14 - 3760563305 x^13 + 3991133690 x^12 - 3450574470 x^11 + 2418714751 x^10 - 1370750375 x^9 + 628081926 x^8 - 228075638 x^7 + 56855445 x^6 - 6423333 x^5 + 4868490 x^4 + 36682 x^3 + 20508 x^2 - 60 x + 4) / (x - 1)^15, {x, 0, 50}], x] (* _Vincenzo Librandi_, May 30 2013 *)

%Y Cf. A172529, A172530, A172531, A172532, A172533.

%K nonn,easy

%O 1,4

%A _Vaclav Kotesovec_, Feb 18 2010