%I #26 Nov 02 2021 06:30:10

%S 0,0,0,24,600,5400,29400,117600,381024,1058400,2613600,5880600,

%T 12269400,24048024,44717400,79497600,135945600,224726400,360561024,

%U 563376600,859685400,1284221400,1881864600,2709885024,3840540000,5364060000

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

%H Andrew Howroyd, <a href="/A179059/b179059.txt">Table of n, a(n) for n = 1..200</a>

%H 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 <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 a(n) = 4! * binomial(n, 4)^2.

%F From _Colin Barker_, Jan 08 2013: (Start)

%F a(n) = (n^2*(-6+11*n-6*n^2+n^3)^2)/24.

%F G.f.: -24*x^4*(x^4 +16*x^3 +36*x^2 +16*x +1) / (x -1)^9.

%F (End)

%F From _Amiram Eldar_, Nov 02 2021: (Start)

%F Sum_{n>=4} 1/a(n) = (20*Pi^2 - 197)/9.

%F Sum_{n>=4} (-1)^n/a(n) = (64*log(2) - 44)/9. (End)

%t 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 a[n_] := If[n<4, 0, Coefficient[n!*LaguerreL[n, x], x, n-4] // Abs];

%t Array[a, 30] (* _Jean-François Alcover_, Jun 14 2018, after A144084 *)

%o (PARI) a(n) = 4! * binomial(n, 4)^2; \\ _Andrew Howroyd_, Feb 13 2018

%Y Column k=4 of A144084.

%Y Cf. A179058 (3 rooks), A179060 (5 rooks).

%K easy,nonn

%O 1,4

%A _Thomas Zaslavsky_, Jun 27 2010