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A334644 a(n) is the total number of down steps between the third and fourth up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

%I A334644 #12 Aug 08 2020 01:42:42
%S A334644 0,0,0,83,299,1263,6076,31307,168561,936161,5321611,30804795,
%T A334644 180939408,1075636912,6459103704,39120216196,238692219923,
%U A334644 1465783144605,9052278085129,56185368932615,350293215459915,2192731008315015,13775745283576920,86831135890324875
%N A334644 a(n) is the total number of down steps between the third and fourth up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.
%C A334644 For n = 3, there is no 4th up step, a(3) = 83 enumerates the total number of down steps between the 3rd up step and the end of the path.
%H A334644 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 A334644 a(0) = a(1) = a(2) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..3}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 30*[n=3] for n > 2, where [ ] is the Iverson bracket.
%o A334644 (SageMath) [binomial(3*n + 1, n)/(3*n + 1) + 4*sum([binomial(3*j + 2,  j) * binomial(3*(n - j), n - j)/(3*j + 2)/(n - j + 1) for j in srange(1, 4)]) - 30*(n==3) if n >= 3 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y A334644 Cf. A001764, A007226, A030983, A334642, A334643.
%K A334644 nonn,easy
%O A334644 0,4
%A A334644 _Benjamin Hackl_, May 12 2020