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%I #19 Apr 18 2022 18:15:41
%S 0,0,0,112,3368,39680,282248,1444928,5865552,20014112,59673360,
%T 159698416,391202680,890095584,1902427800,3853570560,7450556064,
%U 13829016768,24759442464,42930138864,72328779720,118747638592
%N Number of ways to place 5 nonattacking bishops on an n X n board.
%C For any fixed value of k>1, a(n) = n^(2k) /k! - 2n^(2k - 1) /3/(k - 2)! + ...
%H Vincenzo Librandi, <a href="/A172129/b172129.txt">Table of n, a(n) for n = 1..1000</a>
%H Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv preprint arXiv:1609.00853, 2016-2020.
%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_16">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).
%F a(n) = n*(n-2)*(3*n^8 - 34*n^7 + 177*n^6 - 590*n^5 + 1435*n^4 - 2592*n^3 + 3326*n^2 - 2844*n + 1344)/360 if n is even.
%F a(n) = (n-1)*(n-2)*(n-3)*(3*n^7 - 22*n^6 + 80*n^5 - 204*n^4 + 379*n^3 - 464*n^2 + 378*n - 270)/360 if n is odd.
%F G.f.: 8*x^4*(14 + 337*x + 2574*x^2 + 9871*x^3 + 22040*x^4 + 31334*x^5 + 28808*x^6 + 17522*x^7 + 6666*x^8 + 1593*x^9 + 186*x^10 + 15*x^11) / ((1-x)^11*(1+x)^5). - _Vaclav Kotesovec_, Mar 25 2010
%F a(n) = (1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(1-(-1)^n)/2 ). - _G. C. Greubel_, Apr 17 2022
%t CoefficientList[Series[8*x^4*(14 +337*x +2574*x^2 +9871*x^3 +22040*x^4 +31334*x^5 +28808*x^6 +17522*x^7 +6666*x^8 +1593*x^9 +186*x^10 +15*x^11)/((1-x)^11*(1+x)^5), {x, 0, 50}], x]] (* _Vincenzo Librandi_, May 02 2013 *)
%o (SageMath) [(1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(n%2) ) for n in (1..50)] # _G. C. Greubel_, Apr 17 2022
%Y Cf. A108792, A172123, A172124, A172127.
%K nonn,easy
%O 1,4
%A _Vaclav Kotesovec_, Jan 26 2010
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