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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.
7

%I #27 Feb 22 2020 18:01:41

%S 4,114,1492,13992,109538,769632,5050616,31702275,193204684,1154354559,

%T 6805263818,39756392269,230829718918,1334626765852,7694795830792,

%U 44279453377166,254475676808510,1461211112505546,8385454709982584,48102877501302765,275868835046218560

%N 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.

%C 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.)

%H Colin Barker, <a href="/A260575/b260575.txt">Table of n, a(n) for n = 3..1000</a>

%H E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.

%F 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).

%t 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 *)

%Y Column k=3 of A269742.

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

%K nonn,easy

%O 3,1

%A _Steve Butler_, Jul 29 2015