A260727 Number of ways to place 3n rooks on an n X n board, with 3 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

1, 23, 325, 3368, 28819, 218788, 1539399, 10314315, 66953292, 425761614, 2671506918, 16618186770, 102796975770, 633596982417, 3896224129259, 23924104985984, 146764696175937, 899809941054468, 5514653407814317, 33789681789605283, 207007665004469906

Offset

2,2

Comments

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 3 balls when we can catch/throw up to 3 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1, inclusive.)

Links

Colin Barker, Table of n, a(n) for n = 2..1000

E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Formula

G.f.: -(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1).

Mathematica

CoefficientList[Series[-(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/((x^2)*(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 30 2015 *)

Crossrefs

Column k=3 of A269743.

Cf. A260575, A260582, A260583, A260584, A260585.

Sequence in context: A329250 A022747 A270498 * A000543 A220648 A243422

Adjacent sequences: A260724 A260725 A260726 * A260728 A260729 A260730

Keyword

nonn,easy

Author

Chris Cox, Jul 30 2015

Status

approved