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A334979 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_1-Dyck paths of length 4*n. A 3_1-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 = -1.

%I A334979 #20 Aug 25 2025 15:14:15
%S A334979 0,1,16,132,1034,8134,64880,525132,4307512,35750473,299759200,
%T A334979 2535849836,21619615164,185582339740,1602675301920,13915031036412,
%U A334979 121396437548136,1063653520870612,9355905795325888,82585983533819920,731350409249262330,6495673923406863630
%N A334979 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_1-Dyck paths of length 4*n. A 3_1-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 = -1.
%C A334979 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 A334979 Michael De Vlieger, <a href="/A334979/b334979.txt">Table of n, a(n) for n = 0..1027</a>
%H A334979 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 A334979 a(0) = 0 and a(n) = binomial(4*n+6, n+1)/(2*n+3) - 4*binomial(4*n+2, n)/(2*n+1) for n > 0.
%F A334979 G.f.: ((1 - 4*x)*hypergeom([1/2, 3/4, 5/4], [4/3, 5/3], 2^8*x/3^3) - 1 + 2*x)/x. - _Stefano Spezia_, Aug 25 2025
%e A334979 For n = 2, the 3_1-Dyck paths are UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 16.
%t A334979 a[0] = 0; a[n_] := Binomial[4*n+6, n+1]/(2*n + 3) - 4 * Binomial[4*n + 2, n]/(2*n + 1); Array[a, 22, 0]
%o A334979 (SageMath) [binomial(4*n + 6, n + 1)/(2*n + 3) - 4*binomial(4*n + 2, n)/(2*n + 1) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 19 2020
%Y A334979 Cf. A334976, A334977, A334978, A334980.
%K A334979 nonn,easy,changed
%O A334979 0,3
%A A334979 _Sarah Selkirk_, May 18 2020