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%I #11 Aug 07 2020 12:08:00
%S 0,6,31,158,975,6639,48050,362592,2820789,22460120,182141553,
%T 1499143282,12490923757,105150960654,892973346300,7640934031920,
%U 65813450140017,570160918044288,4964875184429660,43431741548248440,381496856026500220,3363457643008999635
%N a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.
%C 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 For n = 1, there is no 2nd up step, a(1) = 6 enumerates the total number of down steps between the 1st up step and the end of the path.
%H 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 a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) + 9*binomial(4*(n-1), n-1)/n - 6*[n=1] for n > 0, where [ ] is the Iverson bracket.
%e For n = 1, the 3_2-Dyck paths are DDUD, DUDD, UDDD. This corresponds to a(1) = 1 + 2 + 3 = 6 down steps between the 1st up step and the end of the path.
%t a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - Binomial[4*n + 2, n]/(n + 1) + 9 * Binomial[4*(n - 1), n - 1]/n - 6 * Boole[n == 1]; Array[a, 22, 0] (* _Amiram Eldar_, May 13 2020 *)
%o (SageMath) [3*binomial(4*n, n)/(n + 1) - binomial(4*n + 2, n)/(n + 1) + 9*binomial(4*(n - 1), n - 1)/n - 6*(n==1) if n > 0 else 0 for n in srange(30)] # _Benjamin Hackl_, May 13 2020
%Y Cf. A001764, A007226, A007228, A334609, A334647, A334648, A334649, A334785.
%K nonn,easy
%O 0,2
%A _Benjamin Hackl_, May 13 2020