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

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Last modified January 30 03:32 EST 2023. Contains 359939 sequences. (Running on oeis4.)