A172967 Number of ways to place 5 nonattacking knights on an n X n cylindrical board.

0, 0, 0, 208, 3210, 58056, 458157, 2524176, 10587591, 36576380, 109008735, 289450344, 700477401, 1570789892, 3304892985, 6586928032, 12530769343, 22891446252

Offset

1,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Formula

Explicit formula: a(n) = n*(n^9-90n^7+120n^6+3395n^5-8160n^4-62130n^3+204000n^2+463464n-1888080)/120, n>=10. For any fixed value of k > 1, a(n) = n^(2k)/k! - 9n^(2k-2)/2/(k-2)! + 6n^(2k-3)/(k-2)! + ... [Vaclav Kotesovec, Jan 31 2010]

G.f.: -x^4*(468*x^16-7964*x^15+57164*x^14-238936*x^13+664383*x^12-1323653*x^11+1986964*x^10-2334676*x^9+2209082*x^8-1718662*x^7+1118210*x^6-595746*x^5+216519*x^4-38229*x^3+34186*x^2+922*x+208)/(x-1)^11. [Vaclav Kotesovec, Mar 25 2010]

Mathematica

CoefficientList[Series[- x^3 (468 x^16 - 7964 x^15 + 57164 x^14 - 238936 x^13 + 664383 x^12 - 1323653 x^11 + 1986964 x^10 - 2334676 x^9 + 2209082 x^8 - 1718662 x^7 + 1118210 x^6 - 595746 x^5 + 216519 x^4 - 38229 x^3 + 34186 x^2 + 922 x + 208) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

Crossrefs

Cf. A172532, A172136, A172964, A172965, A172966.

Sequence in context: A235674 A235444 A255074 * A343507 A231111 A339762

Adjacent sequences: A172964 A172965 A172966 * A172968 A172969 A172970

Keyword

nonn,easy

Author

Vaclav Kotesovec, Feb 06 2010

Status

approved