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
KEYWORD
nonn
AUTHOR
Scarlitte Ponce, Jul 29 2015
STATUS
approved