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

0, 0, 0, 24, 600, 5400, 29400, 117600, 381024, 1058400, 2613600, 5880600, 12269400, 24048024, 44717400, 79497600, 135945600, 224726400, 360561024, 563376600, 859685400, 1284221400, 1881864600, 2709885024, 3840540000, 5364060000

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

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

From Colin Barker, Jan 08 2013: (Start)

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

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

(End)

From Amiram Eldar, Nov 02 2021: (Start)

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

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

Mathematica

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 *)
a[n_] := If[n<4, 0, Coefficient[n!*LaguerreL[n, x], x, n-4] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

Prog

(PARI) a(n) = 4! * binomial(n, 4)^2; \\ Andrew Howroyd, Feb 13 2018

Crossrefs

Column k=4 of A144084.

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

Sequence in context: A041265 A042106 A158637 * A334281 A055359 A154128

Adjacent sequences: A179056 A179057 A179058 * A179060 A179061 A179062

Keyword

easy,nonn

Author

Thomas Zaslavsky, Jun 27 2010

Status

approved