A061998 Number of ways to place 5 nonattacking kings on an n X n board.

0, 0, 0, 0, 0, 1974, 42368, 397014, 2326320, 10087628, 35464464, 106783320, 285336128, 693331146, 1558986816, 3286192514, 6558317232, 12488282352, 22829958032, 40269324564, 68817690624, 114333609854, 185205015936

Offset

0,6

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.

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

Formula

G.f.: 2*x^5*(987 + 10327*x + 19768*x^2 - 18152*x^3 - 2711*x^4 + 5149*x^5 + 1774*x^6 - 2882*x^7 + 958*x^8 - 98*x^9)/(1 - x)^11.

Recurrence: a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11), n >= 15.

Explicit formula (V.Kotesovec, 1992): a(n) = (n - 4)*(n^9 + 4*n^8 - 74*n^7 - 176*n^6 + 2411*n^5 + 1844*n^4 - 38194*n^3 + 18944*n^2 + 236520*n - 316320)/120, n >= 4.

a(n) = A193580(n,5). - R. J. Mathar, Sep 03 2016

Mathematica

CoefficientList[Series[2*x^5*(987 +10327*x +19768*x^2 -18152*x^3 -2711*x^4 +5149*x^5 +1774*x^6 -2882*x^7 +958*x^8 -98*x^9)/(1-x)^11, {x, 0, 45}], x] (* Vincenzo Librandi, May 02 2013 *)

Prog

(SageMath) [0, 0, 0, 0]+[(n-4)*(n^9 +4*n^8 -74*n^7 -176*n^6 +2411*n^5 +1844*n^4 -38194*n^3 +18944*n^2 +236520*n -316320)/120 for n in (4..50)] # G. C. Greubel, May 01 2022

Crossrefs

Cf. A061995, A061996, A061997, A193580.

Sequence in context: A135844 A135845 A121995 * A205653 A206182 A205364

Adjacent sequences: A061995 A061996 A061997 * A061999 A062000 A062001

Keyword

nonn,easy

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Status

approved