A179063 Number of non-attacking placements of 8 rooks on an n X n board.

0, 0, 0, 0, 0, 0, 0, 40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000, 23828156352000, 77203226580480, 230333593351680, 639815537088000, 1669577821632000, 4122835028928000, 9693243245790720, 21809797303029120

Offset

1,8

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 (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Formula

a(n) = 8!*binomial(n,8)^2.

G.f.: -40320*x^8*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x -1)^17. - Colin Barker, Jan 08 2013

From Amiram Eldar, Sep 27 2025: (Start)

Sum_{n>=8} 1/a(n) = 286*Pi^2/315 - 82987349/9261000.

Sum_{n>=8} (-1)^n/a(n) = 2048*log(2)/11025 - 149026/1157625. (End)

Mathematica

a[n_] := 8! * Binomial[n, 8]^2; Array[a, 24] (* Amiram Eldar, Sep 27 2025 *)

Prog

(PARI) a(n) = 8!*binomial(n, 8)^2 \\ Andrew Howroyd, Feb 13 2018

Crossrefs

Column k=8 of A144084.

Cf. A179062 (7 rooks), A179064 (9 rooks).

Sequence in context: A254081 A228911 A213878 * A246196 A246616 A291115

Adjacent sequences: A179060 A179061 A179062 * A179064 A179065 A179066

Keyword

easy,nonn

Author

Thomas Zaslavsky, Jun 27 2010

Status

approved