cp's OEIS Frontend

A084509 Number of ground-state 3-ball juggling sequences of period n.

%I A084509 #55 Apr 25 2022 08:03:57
%S A084509 1,1,2,6,24,96,384,1536,6144,24576,98304,393216,1572864,6291456,
%T A084509 25165824,100663296,402653184,1610612736,6442450944,25769803776,
%U A084509 103079215104,412316860416,1649267441664,6597069766656,26388279066624,105553116266496,422212465065984
%N A084509 Number of ground-state 3-ball juggling sequences of period n.
%C A084509 This sequence counts the length n asynchronic site swaps given in A084501/A084502.
%C A084509 Equals row sums of triangle A145463. - _Gary W. Adamson_, Oct 11 2008
%C A084509 a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4, 1>5} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the first element is the largest. - _Sergey Kitaev_, Dec 11 2020
%C A084509 a(n) is the number of permutations p[1]..p[n] of {1,...,n} with p[j+1] < p[j]+4 for 0 < j < n. - _Don Knuth_, Apr 25 2022
%D A084509 B. Polster, The Mathematics of Juggling, Springer-Verlag, 2003, p. 48.
%H A084509 Michael De Vlieger, <a href="/A084509/b084509.txt">Table of n, a(n) for n = 0..1662</a>
%H A084509 Fan Chung and R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Amer. Math. Monthly 115(3) (2008), 185-19.
%H A084509 Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A084509 Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H A084509 Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, <a href="https://arxiv.org/abs/2101.12061">Permutations Avoiding Certain Partially-ordered Patterns</a>, arXiv:2101.12061 [math.CO], 2021.
%H A084509 <a href="/index/J#Juggling">Index entries for sequences related to juggling</a>
%H A084509 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F A084509 a(n) = n! for n <= 4, a(n) = 6*4^(n-3) = A002023(n-3) for n >= 3.
%F A084509 G.f.: 1 + x*(1 - 2*x - 2*x^2)/(1 - 4*x). - _Philippe Deléham_, Aug 16 2005
%p A084509 A084509 := n -> `if`((n<4),n!,6*(4^(n-3)));
%p A084509 INVERT([seq(A084519(n),n=1..12)]);
%t A084509 LinearRecurrence[{4},{1,2,6},30] (* _Harvey P. Dale_, Aug 23 2018 *)
%Y A084509 First differences of A084508.
%Y A084509 INVERT transform of A084519.
%Y A084509 Cf. A002023, A084501, A084502, A084529, A145463.
%K A084509 nonn,easy
%O A084509 0,3
%A A084509 _Antti Karttunen_, Jun 02 2003
%E A084509 a(0)=1 prepended by _Alois P. Heinz_, Dec 11 2020