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A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.

%I A334647 #18 Aug 08 2020 01:39:55
%S A334647 0,5,16,78,470,3153,22588,169188,1308762,10374460,83829856,687929086,
%T A334647 5717602930,48030047206,407142435000,3478286028840,29917720938690,
%U A334647 258866494630164,2251694583485824,19677972159742360,172694287830500440,1521328368800877065
%N A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.
%C A334647 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 A334647 For n = 1, there is no 2nd up step, a(1) = 5 enumerates the total number of down steps between the 1st up step and the end of the path.
%H A334647 Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down-step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%F A334647 a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.
%e A334647 For n = 1, the 3_1-Dyck paths are UDDD, DUDD. This corresponds to a(1) = 3 + 2 = 5 down steps between the 1st up step and the end of the path.
%e A334647 For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 = 16 down steps between the 1st and 2nd up step.
%t A334647 a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - 2 * Binomial[4*n + 1, n]/(n + 1) + 6 * Binomial[4*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 22, 0] (* _Amiram Eldar_, May 12 2020 *)
%o A334647 (SageMath) [3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y A334647 Cf. A002293, A007226, A007228, A334645, A334646, A334648, A334649, A334680, A334682, A334785.
%K A334647 nonn,easy
%O A334647 0,2
%A A334647 _Benjamin Hackl_, May 12 2020