A260575 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

4, 114, 1492, 13992, 109538, 769632, 5050616, 31702275, 193204684, 1154354559, 6805263818, 39756392269, 230829718918, 1334626765852, 7694795830792, 44279453377166, 254475676808510, 1461211112505546, 8385454709982584, 48102877501302765, 275868835046218560

Offset

3,1

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 2 balls at a time. (Minimal in the sense that the throws are between 0 and n-1.)

Links

Colin Barker, Table of n, a(n) for n = 3..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.: -(625*x^12 - 2206*x^11 + 4397*x^10 - 6648*x^9 + 7058*x^8 - 4674*x^7 + 1748*x^6 - 300*x^5 + 2*x^4 + 4*x^3)/(2600*x^13 - 21100*x^12 + 77590*x^11 - 171025*x^10 + 251874*x^9 - 261466*x^8 + 196626*x^7 - 108337*x^6 + 43682*x^5 - 12713*x^4 + 2592*x^3 - 350*x^2 + 28*x - 1).

Mathematica

Rest[Rest[Rest[CoefficientList[Series[-(625 x^12 - 2206 x^11 + 4397 x^10 - 6648 x^9 + 7058 x^8 - 4674 x^7 + 1748 x^6 - 300 x^5 + 2 x^4 + 4 x^3)/(2600 x^13 - 21100 x^12 + 77590 x^11 - 171025 x^10 + 251874 x^9 - 261466 x^8 + 196626 x^7 - 108337 x^6 + 43682 x^5 - 12713 x^4 + 2592 x^3 - 350 x^2 + 28 x - 1), {x, 0, 33}], x]]]] (* Vincenzo Librandi, Jul 30 2015 *)

Crossrefs

Column k=3 of A269742.

Cf. A260582, A260583, A260584, A260585, A260727.

Sequence in context: A300036 A199834 A194495 * A275747 A239942 A261457

Adjacent sequences: A260572 A260573 A260574 * A260576 A260577 A260578

Keyword

nonn,easy

Author

Steve Butler, Jul 29 2015

Status

approved