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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.

%I A334785 #19 Oct 23 2022 01:21:48
%S A334785 0,3,13,74,480,3363,24794,189540,1488744,11941820,97412601,805602850,
%T A334785 6738919408,56918898330,484750343700,4158094853640,35891774969112,
%U A334785 311529010178628,2717299393716836,23806014817182600,209389427777770240,1848322153489496355
%N 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.
%H A334785 Stefano Spezia, <a href="/A334785/b334785.txt">Table of n, a(n) for n = 0..1000</a>
%H A334785 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 A334785 a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) for n > 0.
%F A334785 a(n) ~ c*2^(8*n)*3^(-3*n)/n^(3/2), where c = (11/9)*sqrt(2/(3*Pi)). - _Stefano Spezia_, Oct 19 2022
%e A334785 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.
%t A334785 a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n+1) - Binomial[4*n+2, n]/(n+1); Array[a, 22, 0]
%Y A334785 Cf. A001764, A002293, A002294, A334786, A334787.
%K A334785 nonn,easy
%O A334785 0,2
%A A334785 _Sarah Selkirk_, May 11 2020