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A334787 a(n) is the total number of down steps before the first up step in all 4_3-Dyck paths of length 5*n. A 4_3-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 = -3.

%I A334787 #18 Oct 23 2022 01:22:09
%S A334787 0,6,34,251,2105,19040,181076,1784728,18067803,186754590,1962728460,
%T A334787 20910164730,225308533359,2451112021568,26885549373440,
%U A334787 297008527319440,3301615350645935,36903975448964670,414518195957729886,4676429192392769805,52965796433899543810
%N A334787 a(n) is the total number of down steps before the first up step in all 4_3-Dyck paths of length 5*n. A 4_3-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 = -3.
%H A334787 Stefano Spezia, <a href="/A334787/b334787.txt">Table of n, a(n) for n = 0..900</a>
%H A334787 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 A334787 a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - binomial(5*n+3, n)/(n+1) for n > 0.
%F A334787 a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (131/128)*sqrt(5/(2*Pi)). - _Stefano Spezia_, Oct 19 2022
%e A334787 For n = 1, there are the 4_3-Dyck paths UDDDD, DUDDD, DDUDD, DDDUD. Before the first up step there are a(1) = 0 + 1 + 2 + 3 = 6 down steps in total.
%t A334787 a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - Binomial[5*n+3, n]/(n+1); Array[a, 21, 0]
%Y A334787 Cf. A001764, A002293, A002294, A334785, A334786.
%K A334787 nonn,easy
%O A334787 0,2
%A A334787 _Sarah Selkirk_, May 11 2020