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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.
8
1, 11, 72, 367, 1630, 6680, 26082, 98870, 368045, 1354850, 4953503, 18035279, 65499031, 237511321, 860471110, 3115667369, 11277816388, 40814611818, 147692103728, 534404499040, 1933597628291, 6996040095316, 25312367524557, 91581960107817, 331348634005165
OFFSET
2,2
COMMENTS
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.)
LINKS
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
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.
Index entries for linear recurrences with constant coefficients, signature (12,-59,155,-236,209,-100,20).
FORMULA
G.f.: -(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).
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Column k=2 of A269742.
Sequence in context: A156149 A258402 A006975 * A084900 A300968 A218619
KEYWORD
nonn
AUTHOR
Jeffrey Davis, Jul 29 2015
STATUS
approved