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A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

%I A260585 #41 Aug 26 2025 07:26:44
%S A260585 1,11,72,367,1630,6680,26082,98870,368045,1354850,4953503,18035279,
%T A260585 65499031,237511321,860471110,3115667369,11277816388,40814611818,
%U A260585 147692103728,534404499040,1933597628291,6996040095316,25312367524557,91581960107817,331348634005165
%N A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.
%C A260585 a(n) is the number of minimal multiplex juggling patterns of period n using exactly 2 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1.)
%H A260585 Colin Barker, <a href="/A260585/b260585.txt">Table of n, a(n) for n = 2..1000</a>
%H A260585 Esther M. Banaian, <a href="http://digitalcommons.csbsju.edu/honors_thesis/24">Generalized Eulerian Numbers and Multiplex Juggling Sequences</a>, (2016). All College Thesis Program. Paper 24.
%H A260585 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.
%H A260585 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (12,-59,155,-236,209,-100,20).
%F A260585 G.f.: -x^2*(5*x^4-3*x^3-x^2-x+1)/((1-5*x+5*x^2)*(2*x-1)^2*(x-1)^3).
%F A260585 a(n) = 12*a(n-1) - 59*a(n-2) + 155*a(n-3) - 236*a(n-4) + 209*a(n-5) - 100*a(n-6) + 20*a(n-7). - _Wesley Ivan Hurt_, Jan 01 2024
%F A260585 a(n) = (n+2)*(n-1)/2-2^n*(1+3*n/2)+2*A030191(n)-5*A030191(n-1). - _R. J. Mathar_, Aug 26 2025
%t A260585 CoefficientList[Series[-(5*x^4 - 3*x^3 - x^2 - x + 1)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Aug 16 2015 *)
%o A260585 (PARI) Vec(-(5*x^6 - 3*x^5 - x^4 - x^3 + x^2)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1) + O(x^40)) \\ _Michel Marcus_, Aug 17 2015
%Y A260585 Column k=2 of A269742.
%Y A260585 Cf. A260575, A260582, A260583, A260584, A260727.
%K A260585 nonn,easy,changed
%O A260585 2,2
%A A260585 _Jeffrey Davis_, Jul 29 2015