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%I #12 Aug 08 2020 01:42:59
%S 0,1,9,53,299,1692,9690,56221,330165,1959945,11745435,70974252,
%T 432019844,2646716264,16307880462,100996570221,628356589721,
%U 3925544432355,24616047166095,154886752443885,977595783524955,6187863825170160,39269844955755960,249819662230403148
%N 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.
%C 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.
%H A. Asinowski, B. Hackl, and S. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%F 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.
%e 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.
%t a[0] = 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]
%o (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
%Y Cf. A334976, A334978, A334979, A334980.
%K nonn,easy
%O 0,3
%A _Sarah Selkirk_, May 18 2020
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