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%I #21 May 02 2022 01:39:24
%S 0,0,0,0,0,1974,42368,397014,2326320,10087628,35464464,106783320,
%T 285336128,693331146,1558986816,3286192514,6558317232,12488282352,
%U 22829958032,40269324564,68817690624,114333609854,185205015936
%N Number of ways to place 5 nonattacking kings on an n X n board.
%H Vincenzo Librandi, <a href="/A061998/b061998.txt">Table of n, a(n) for n = 0..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>, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
%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 G.f.: 2*x^5*(987 + 10327*x + 19768*x^2 - 18152*x^3 - 2711*x^4 + 5149*x^5 + 1774*x^6 - 2882*x^7 + 958*x^8 - 98*x^9)/(1 - x)^11.
%F Recurrence: a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11), n >= 15.
%F Explicit formula (V.Kotesovec, 1992): a(n) = (n - 4)*(n^9 + 4*n^8 - 74*n^7 - 176*n^6 + 2411*n^5 + 1844*n^4 - 38194*n^3 + 18944*n^2 + 236520*n - 316320)/120, n >= 4.
%F a(n) = A193580(n,5). - _R. J. Mathar_, Sep 03 2016
%t CoefficientList[Series[2*x^5*(987 +10327*x +19768*x^2 -18152*x^3 -2711*x^4 +5149*x^5 +1774*x^6 -2882*x^7 +958*x^8 -98*x^9)/(1-x)^11, {x,0,45}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%o (SageMath) [0,0,0,0]+[(n-4)*(n^9 +4*n^8 -74*n^7 -176*n^6 +2411*n^5 +1844*n^4 -38194*n^3 +18944*n^2 +236520*n -316320)/120 for n in (4..50)] # _G. C. Greubel_, May 01 2022
%Y Cf. A061995, A061996, A061997, A193580.
%K nonn,easy
%O 0,6
%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001
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