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%I #18 Sep 08 2022 08:45:50
%S 0,0,8,260,2728,16428,70792,242856,706048,1809464,4199064,8992684,
%T 18024072,34170724,61784632,107243472,179645376,291667440,460615272,
%U 709686228,1069477928,1579767068,2291594536,3269684088,4595235136
%N Number of ways to place 4 nonattacking bishops on an n X n board.
%D E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
%H Vincenzo Librandi, <a href="/A172127/b172127.txt">Table of n, a(n) for n = 1..1000</a>
%H Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv:1609.00853 [math.CO], 2016.
%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>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1).
%F Explicit formula (Karl Fabel, 1966): a(n) = n(n - 2)(15n^6 - 90n^5 + 260n^4 - 524n^3 + 727n^2 - 646n + 348)/360 if n is even and a(n) = (n - 1)(n - 2)(15n^6 - 75n^5 + 185n^4 - 339n^3 + 388n^2 - 258n + 180)/360 if n is odd.
%F G.f.: 4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3). - _Vaclav Kotesovec_, Mar 25 2010
%t CoefficientList[Series[-4 x^2 (6 x^8 + 57 x^7 + 316 x^6 + 763 x^5 + 1056 x^4 + 791 x^3 + 316 x^2 + 53 x + 2) / ((x-1)^9 (x+1)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%t LinearRecurrence[{6,-12,2,27,-36,0,36,-27,-2,12,-6,1},{0,0,8,260,2728,16428,70792,242856,706048,1809464,4199064,8992684},30] (* _Harvey P. Dale_, Dec 09 2017 *)
%o (PARI) x='x+O('x^50); concat([0,0], Vec(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))) \\ _G. C. Greubel_, Nov 04 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))); // _G. C. Greubel_, Nov 04 2018
%Y Cf. A061994, A172123, A172124.
%K nonn,easy
%O 1,3
%A _Vaclav Kotesovec_, Jan 26 2010
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