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A334644
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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.
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1
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0, 0, 0, 83, 299, 1263, 6076, 31307, 168561, 936161, 5321611, 30804795, 180939408, 1075636912, 6459103704, 39120216196, 238692219923, 1465783144605, 9052278085129, 56185368932615, 350293215459915, 2192731008315015, 13775745283576920, 86831135890324875
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..23.
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
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FORMULA
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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.
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PROG
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(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
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CROSSREFS
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Cf. A001764, A007226, A030983, A334642, A334643.
Sequence in context: A244776 A251082 A141570 * A288172 A341338 A031433
Adjacent sequences: A334641 A334642 A334643 * A334645 A334646 A334647
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KEYWORD
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nonn,easy
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AUTHOR
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Benjamin Hackl, May 12 2020
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STATUS
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approved
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