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%I #20 Apr 29 2022 03:49:04
%S 0,0,0,8,140,964,3920,11860,29708,65240,129984,240240,418220,693308,
%T 1103440,1696604,2532460,3684080,5239808,7305240,10005324,13486580,
%U 17919440,23500708,30456140,39043144,49553600,62316800
%N Number of ways to place 3 nonattacking kings on an n X n board.
%H Vincenzo Librandi, <a href="/A061996/b061996.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Placing non-attacking queens and kings on boards of various sizes</a>, part of "Between chessboard and computer", 1996, pp. 204 - 206.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: 4*x^3*(2 + 21*x + 38*x^2 - 42*x^3 + 11*x^4)/(1 - x)^7.
%F Recurrence: a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), n >= 8.
%F a(n) = (n-1)*(n-2)*(n^4 + 3*n^3 - 20*n^2 - 30*n + 132)/6, n >= 1.
%F a(n) = A193580(n,3). - _R. J. Mathar_, Sep 03 2016
%F E.g.f.: -44 + (1/6)*(264 -264*x +132*x^2 -36*x^3 +38*x^4 +15*x^5 +x^6)*exp(x). - _G. C. Greubel_, Apr 29 2022
%t CoefficientList[Series[4x^3(2 +21x +38x^2 -42x^3 +11x^4)/(1-x)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%o (SageMath) [(n-1)*(n-2)*(n^4+3*n^3-20*n^2-30*n+132)/6 -44*bool(n==0) for n in (0..40)] # _G. C. Greubel_, Apr 29 2022
%Y Cf. A061995, A061997, A061998, A193580.
%K nonn,easy
%O 0,4
%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001
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