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%I #12 Aug 07 2020 12:09:07
%S 0,0,6,52,409,3208,25484,205452,1679332,13894848,116193246,980658172,
%T 8343605534,71492410640,616418176920,5344364518140,46565472754044,
%U 407529832131712,3580911446989368,31579384975219920,279414033129153065,2479725948121016040
%N a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.
%C For n = 1, there is no (n-1)-th up step, a(1) = 0 is the total number of down steps before the first up step.
%H A. Asinowski, B. Hackl, and S. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%F a(0) = 0 and a(n) = binomial(4*n+5, n+1)/(4*n+5) - 4*binomial(4*n+1, n)/(4*n+1) for n > 0.
%e For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. Therefore, the total number of down steps between the first and second up steps is a(2) = 3 + 2 + 1 + 0 = 6.
%t a[0] = 0; a[n_] := Binomial[4*n+5, n+1]/(4*n + 5) - 4 * Binomial[4*n + 1, n]/(4*n + 1); Array[a, 22, 0]
%o (SageMath) [binomial(4*n + 5, n + 1)/(4*n + 5) - 4*binomial(4*n + 1, n)/(4*n + 1) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 19 2020
%Y Cf. A334976, A334977, A334979, A334980.
%K nonn,easy
%O 0,3
%A _Sarah Selkirk_, May 18 2020
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