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A334786 a(n) is the total number of down steps before the first up step in all 4_2-Dyck paths of length 5*n. A 4_2-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

%I A334786 #18 Oct 23 2022 01:21:59
%S A334786 0,3,16,115,950,8510,80388,788392,7950930,81935425,859005840,
%T A334786 9132977490,98240702586,1067197649840,11691092372000,129011823098160,
%U A334786 1432744619523530,16000911127589355,179590878292003200,2024687100104286525,22917687021180660940
%N A334786 a(n) is the total number of down steps before the first up step in all 4_2-Dyck paths of length 5*n. A 4_2-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.
%H A334786 Stefano Spezia, <a href="/A334786/b334786.txt">Table of n, a(n) for n = 0..900</a>
%H A334786 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 A334786 a(0) = 0 and a(n) = 4 * binomial(5*n, n)/(n+1) - 2 * binomial(5*n+2, n)/(n+1) for n > 0.
%F A334786 a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (7/16)*sqrt(5/(2*Pi)). - _Stefano Spezia_, Oct 19 2022
%e A334786 For n = 1, there are the 4_2-Dyck paths UDDDD, DUDDD, DDUDD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.
%t A334786 a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - 2 * Binomial[5*n+2, n]/(n+1); Array[a, 21, 0]
%Y A334786 Cf. A001764, A002293, A002294, A334785, A334787.
%K A334786 nonn,easy
%O A334786 0,2
%A A334786 _Sarah Selkirk_, May 11 2020