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 A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n. 4
 0, 0, 0, 236, 1034, 6094, 40996, 295740, 2231022, 17370163, 138473536, 1124433142, 9266859394, 77307427741, 651540030688, 5538977450256, 47442103851930, 409000732566399, 3546232676711824, 30903652601552272, 270529448396053576, 2377829916885541565 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. For n = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path. LINKS Andrei Asinowski, Benjamin Hackl, Sarah J. 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(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 52*[n=3] for n > 2, where [ ] is the Iverson bracket. PROG (SageMath) [binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 4)]) - 52*(n==3) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020 CROSSREFS Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785. Sequence in context: A138819 A273181 A259639 * A233983 A251491 A251484 Adjacent sequences:  A334646 A334647 A334648 * A334650 A334651 A334652 KEYWORD nonn,easy AUTHOR Benjamin Hackl, May 12 2020 STATUS approved

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Last modified October 17 13:32 EDT 2021. Contains 348049 sequences. (Running on oeis4.)