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A179059 Number of non-attacking placements of 4 rooks on an n X n board.

%I A179059 #26 Nov 02 2021 06:30:10
%S A179059 0,0,0,24,600,5400,29400,117600,381024,1058400,2613600,5880600,
%T A179059 12269400,24048024,44717400,79497600,135945600,224726400,360561024,
%U A179059 563376600,859685400,1284221400,1881864600,2709885024,3840540000,5364060000
%N A179059 Number of non-attacking placements of 4 rooks on an n X n board.
%H A179059 Andrew Howroyd, <a href="/A179059/b179059.txt">Table of n, a(n) for n = 1..200</a>
%H A179059 Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, <a href="https://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
%H A179059 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A179059 a(n) = 4! * binomial(n, 4)^2.
%F A179059 From _Colin Barker_, Jan 08 2013: (Start)
%F A179059 a(n) = (n^2*(-6+11*n-6*n^2+n^3)^2)/24.
%F A179059 G.f.: -24*x^4*(x^4 +16*x^3 +36*x^2 +16*x +1) / (x -1)^9.
%F A179059 (End)
%F A179059 From _Amiram Eldar_, Nov 02 2021: (Start)
%F A179059 Sum_{n>=4} 1/a(n) = (20*Pi^2 - 197)/9.
%F A179059 Sum_{n>=4} (-1)^n/a(n) = (64*log(2) - 44)/9. (End)
%t A179059 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,0,24,600,5400,29400,117600,381024},40] (* _Harvey P. Dale_, Feb 19 2013 *)
%t A179059 a[n_] := If[n<4, 0, Coefficient[n!*LaguerreL[n, x], x, n-4] // Abs];
%t A179059 Array[a, 30] (* _Jean-François Alcover_, Jun 14 2018, after A144084 *)
%o A179059 (PARI) a(n) = 4! * binomial(n, 4)^2; \\ _Andrew Howroyd_, Feb 13 2018
%Y A179059 Column k=4 of A144084.
%Y A179059 Cf. A179058 (3 rooks), A179060 (5 rooks).
%K A179059 easy,nonn
%O A179059 1,4
%A A179059 _Thomas Zaslavsky_, Jun 27 2010