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 A334785 a(n) is the total number of down steps before the first up step in all 3_2-Dyck paths of length 4*n. A 3_2-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 = -2. 10
 0, 3, 13, 74, 480, 3363, 24794, 189540, 1488744, 11941820, 97412601, 805602850, 6738919408, 56918898330, 484750343700, 4158094853640, 35891774969112, 311529010178628, 2717299393716836, 23806014817182600, 209389427777770240, 1848322153489496355 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020. FORMULA a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) for n > 0. EXAMPLE For n = 1, there are the 3_2-Dyck paths UDDD, DUDD, DDUD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total. MATHEMATICA a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n+1) - Binomial[4*n+2, n]/(n+1); Array[a, 22, 0] CROSSREFS Cf. A001764, A002293, A002294, A334786, A334787. Sequence in context: A119013 A190878 A156154 * A020094 A189886 A333890 Adjacent sequences:  A334782 A334783 A334784 * A334786 A334787 A334788 KEYWORD nonn,easy AUTHOR Sarah Selkirk, May 11 2020 STATUS approved

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Last modified January 16 06:48 EST 2021. Contains 340204 sequences. (Running on oeis4.)