A179060 Number of non-attacking placements of 5 rooks on an n X n board.

0, 0, 0, 0, 120, 4320, 52920, 376320, 1905120, 7620480, 25613280, 75271680, 198764280, 480960480, 1082161080, 2289530880, 4594961280, 8809274880, 16225246080, 28844881920, 49689816120, 83217546720, 135870624120, 216790801920, 338735628000, 519241008000, 782079948000

Offset

1,5

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. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders, arXiv:1609.00853 [math.CO], 2016-2020.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = 5! * binomial(n, 5)^2.

G.f.: -120*x^5*(x+1)*(x^4+24*x^3+76*x^2+24*x+1) / (x-1)^11. - Colin Barker, Jan 08 2013

From Amiram Eldar, Sep 27 2025: (Start)

Sum_{n>=5} 1/a(n) = 175*Pi^2/72 - 27625/1152.

Sum_{n>=5} (-1)^(n+1)/a(n) = 5*Pi^2/48 - 1175/1152. (End)

Mathematica

a[n_] := If[n<5, 0, Coefficient[n!*LaguerreL[n, x], x, n-5] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)
a[n_] := 5! * Binomial[n, 5]^2; Array[a, 27] (* Amiram Eldar, Sep 27 2025 *)

Prog

(PARI) a(n) = 5! * binomial(n, 5)^2 \\ Andrew Howroyd, Feb 13 2018

Crossrefs

Column k=5 of A144084.

Cf. A179059 (4 rooks), A179061 (6 rooks).

Sequence in context: A139389 A166596 A000514 * A342073 A055360 A001807

Adjacent sequences: A179057 A179058 A179059 * A179061 A179062 A179063

Keyword

easy,nonn

Author

Thomas Zaslavsky, Jun 27 2010

Status

approved