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%I #18 Apr 23 2022 18:38:17
%S 0,0,2,340,9386,97580,649476,3184708,12472084,41199404,119171110,
%T 309957412,739123094,1639655452,3422020324,6778432292,12833460256,
%U 23356032940,41051290730,69954580804
%N Number of ways to place 5 nonattacking knights on an n X n board.
%C For any fixed value of k>1, a(n) = n^(2*k) /k! - 9*n^(2*k - 2) /2/(k - 2)! + 12*n^(2*k - 3) /(k - 2)! + ...
%H Vincenzo Librandi, <a href="/A172136/b172136.txt">Table of n, a(n) for n = 1..1000</a>
%H Vaclav 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>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F Explicit formula: a(n) = (n^10 - 90*n^8 + 240*n^7 + 3235*n^6 - 16320*n^5 - 40530*n^4 + 396480*n^3 - 231656*n^2 - 3359520*n + 6509280)/120, n >= 8.
%F G.f.: 2*x^3 * (74*x^15 -518*x^14 +1110*x^13 +1046*x^12 -11332*x^11 +29950*x^10 -42430*x^9 +32476*x^8 -11684*x^7 -1000*x^6 +15021*x^5 -18443*x^4 -6352*x^3 -2878*x^2 -159*x -1) / (x-1)^11. [_Vaclav Kotesovec_, Mar 25 2010]
%t CoefficientList[Series[2x^2(74 x^15 -518x^14 +1110x^13 +1046x^12 -11332x^11 + 29950x^10 -42430x^9 +32476x^8 -11684x^7 -1000x^6 +15021x^5 -18443x^4 -6352x^3 - 2878x^2 -159x -1)/(x-1)^11, {x,0,40}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%o (SageMath) [0,0,2,340,9386,97580,649476] + [(n^10 -90*n^8 +240*n^7 +3235*n^6 - 16320*n^5 -40530*n^4 +396480*n^3 -231656*n^2 -3359520*n +6509280)/120 for n in (8..50)] # _G. C. Greubel_, Apr 19 2022
%Y Cf. A108792, A172129, A172132, A172134, A172135.
%Y Column k=5 of A244081.
%K nonn,easy
%O 1,3
%A _Vaclav Kotesovec_, Jan 26 2010
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