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A334980 a(n) is the total number of down steps between the (n-1)-th and n-th up steps 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 A334980 #17 Oct 06 2020 02:20:40
%S A334980 0,3,31,248,1941,15334,122915,999456,8231740,68562887,576661761,
%T A334980 4891506968,41801697070,359574305580,3111012673755,27055673506128,
%U A334980 236387476114548,2073957836402524,18264689865840284,161403223665821280,1430768729986730685,12719497076318052990
%N A334980 a(n) is the total number of down steps between the (n-1)-th and n-th up steps 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.
%C A334980 For n = 1, there is no (n-1)-th up step, a(1) = 3 is the total number of down steps before the first up step.
%H A334980 Michael De Vlieger, <a href="/A334980/b334980.txt">Table of n, a(n) for n = 0..1027</a>
%H A334980 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 A334980 a(0) = 0 and a(n) = 3*binomial(4*n+7, n+1)/(4*n+7) - 12*binomial(4*n+3, n)/(4*n+3) for n > 0.
%e A334980 For n = 2, the 3_2-Dyck paths are UDDDDDUD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDDUD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, DDUDDDUD, DDUDDUDD, DDUDUDDD, DDUUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 5 + 4 + 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 31.
%t A334980 a[0] = 0; a[n_] := 3*Binomial[4*n+7, n+1]/(4*n + 7) - 12 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 22, 0]
%o A334980 (SageMath) [3*binomial(4*n + 7, n + 1)/(4*n + 7) - 12*binomial(4*n + 3, n)/(4*n + 3) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 19 2020
%Y A334980 Cf. A334976, A334977, A334978, A334979.
%K A334980 nonn,easy
%O A334980 0,2
%A A334980 _Sarah Selkirk_, May 18 2020