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 A334977 a(n) is the total number of down steps between the (n-1)-th and n-th 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. 4
 0, 1, 9, 53, 299, 1692, 9690, 56221, 330165, 1959945, 11745435, 70974252, 432019844, 2646716264, 16307880462, 100996570221, 628356589721, 3925544432355, 24616047166095, 154886752443885, 977595783524955, 6187863825170160, 39269844955755960, 249819662230403148 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For n = 1, there is no (n-1)-th up step, a(1) = 1 is the total number of down steps before the first up step. 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) = 2*binomial(3*n+5, n+1)/(3*n+5) - 6*binomial(3*n+2, n)/(3*n+2) for n > 0. EXAMPLE For n = 2, the 2_1-Dyck paths are UDDDUD, UDDUDD, UDUDDD, UUDDDD, DUDDUD, DUDUDD, DUUDDD. Therefore the total number of down steps between the first and second up step is a(2) = 3 + 2 + 1 + 0 + 2 + 1 +0 = 9. MATHEMATICA a = 0; a[n_] := 2*Binomial[3*n+5, n+1]/(3*n + 5) - 6 * Binomial[3*n + 2, n]/(3*n + 2); Array[a, 24, 0] PROG (SageMath) [2*binomial(3*n + 5, n + 1)/(3*n + 5) - 6*binomial(3*n + 2, n)/(3*n + 2) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020 CROSSREFS Cf. A334976, A334978, A334979, A334980. Sequence in context: A122588 A277999 A295203 * A038761 A003698 A001688 Adjacent sequences:  A334974 A334975 A334976 * A334978 A334979 A334980 KEYWORD nonn,easy AUTHOR Sarah Selkirk, May 18 2020 STATUS approved

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Last modified January 25 03:42 EST 2022. Contains 350565 sequences. (Running on oeis4.)