A260582 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 4 rooks below the main diagonal.

1, 72, 2438, 48965, 727982, 9002669, 98831244, 1001534339, 9604385112, 88600727292, 795108048465, 6995452987296, 60672964077315, 520801298224219, 4436874672072459, 37592602817393616, 317246106027904761, 2669508900483670024, 22415690107381454687

Offset

3,2

Comments

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that the number of throws is 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.: -(4584825*x^21 - 32402639*x^20 + 116197885*x^19 - 276109240*x^18 + 466638686*x^17 - 565360388*x^16 + 479478061*x^15 - 263101580*x^14 + 64737485*x^13 + 27721713*x^12 - 36043190*x^11 + 18319939*x^10 - 5637417*x^9 + 1094626*x^8 - 124221*x^7 + 5875*x^6 + 171*x^5 - 14*x^4 - x^3)/(19266000*x^22 - 234624000*x^21 + 1345258600*x^20 - 4829459800*x^19 + 12177772645*x^18 - 22934336190*x^17 + 33487611783*x^16 - 38844575208*x^15 + 36384232939*x^14 - 27820436326*x^13 + 17485343731*x^12 - 9066508172*x^11 + 3881838842*x^10 - 1369857572*x^9 + 396588486*x^8 - 93458208*x^7 + 17715207*x^6 - 2654590*x^5 + 306583*x^4 - 26260*x^3 + 1567*x^2 - 58*x + 1).

Mathematica

CoefficientList[Series[-(4584825 x^18 - 32402639 x^17 + 116197885 x^16 - 276109240 x^15 + 466638686 x^14 - 565360388 x^13 + 479478061 x^12 - 263101580 x^11 + 64737485 x^10 + 27721713 x^9 - 36043190 x^8 + 18319939 x^7 - 5637417 x^6 + 1094626 x^5 - 124221 x^4 + 5875 x^3 + 171 x^2 - 14 x - 1)/(19266000 x^22 - 234624000 x^21 + 1345258600 x^20 - 4829459800 x^19 + 12177772645 x^18 - 22934336190  x^17 + 33487611783 x^16 - 38844575208 x^15 + 36384232939 x^14 - 27820436326 x^13 + 17485343731 x^12 - 9066508172 x^11 + 3881838842 x^10 - 1369857572 x^9 + 396588486 x^8 - 93458208 x^7 + 17715207 x^6 - 2654590 x^5 + 306583 x^4 - 26260 x^3 + 1567 x^2 - 58 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 19 2015 *)

Crossrefs

Column k=4 of A269742.

Cf. A260575, A260583, A260584, A260585, A260727.

Sequence in context: A277532 A341556 A014704 * A035118 A017788 A035731

Adjacent sequences: A260579 A260580 A260581 * A260583 A260584 A260585

Keyword

nonn

Author

Scarlitte Ponce, Jul 29 2015

Status

approved