A172136 Number of ways to place 5 nonattacking knights on an n X n board.

0, 0, 2, 340, 9386, 97580, 649476, 3184708, 12472084, 41199404, 119171110, 309957412, 739123094, 1639655452, 3422020324, 6778432292, 12833460256, 23356032940, 41051290730, 69954580804

Offset: 1

Author

Vaclav Kotesovec, Jan 26 2010

Keywords

• nonn
• easy

Comments

For any fixed value of k>1, a(n) = n^(2*k) /k! - 9*n^(2*k - 2) /2/(k - 2)! + 12*n^(2*k - 3) /(k - 2)! + ...

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A108792, A172129, A172132, A172134, A172135.

Column k=5 of A244081.

Programs

Mathematica

CoefficientList[Series[2x^2(74 x^15 -518x^14 +1110x^13 +1046x^12 -11332x^11 + 29950x^10 -42430x^9 +32476x^8 -11684x^7 -1000x^6 +15021x^5 -18443x^4 -6352x^3 - 2878x^2 -159x -1)/(x-1)^11, {x,0,40}], x] (* Vincenzo Librandi, May 02 2013 *)

SageMath

[0,0,2,340,9386,97580,649476] + [(n^10 -90*n^8 +240*n^7 +3235*n^6 - 16320*n^5 -40530*n^4 +396480*n^3 -231656*n^2 -3359520*n +6509280)/120 for n in (8..50)] # G. C. Greubel, Apr 19 2022

Formula

Explicit formula: a(n) = (n^10 - 90*n^8 + 240*n^7 + 3235*n^6 - 16320*n^5 - 40530*n^4 + 396480*n^3 - 231656*n^2 - 3359520*n + 6509280)/120, n >= 8.

G.f.: 2*x^3 * (74*x^15 -518*x^14 +1110*x^13 +1046*x^12 -11332*x^11 +29950*x^10 -42430*x^9 +32476*x^8 -11684*x^7 -1000*x^6 +15021*x^5 -18443*x^4 -6352*x^3 -2878*x^2 -159*x -1) / (x-1)^11. [Vaclav Kotesovec, Mar 25 2010]