A334645 - OEIS

%I #12 Aug 07 2020 12:07:46

%S 0,0,18,52,277,1752,12120,88692,674751,5282160,42267384,344152080,

%T 2842055359,23746693240,200383750632,1705243729560,14617677294675,

%U 126106202849760,1094034474058488,9538676631305712,83536778390997780,734521734171474400,6481894477750488160

%N a(n) is the total number of down steps between the 2nd and 3rd up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

%C For n = 2, there is no 3rd up step, a(2) = 18 enumerates the total number of down steps between the 2nd 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) = a(1) = 0 and a(n) = 3*Sum_{j=0..2} binomial(4*j+1, j) * binomial(4*(n-j), n-j)/((4*j+1) * (n-j+1)) for n > 1.

%e For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 4 + 5 + 6 = 18 down steps between the 2nd up step and the end of the path.

%o (SageMath) [3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 3)]) if n > 1 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020

%Y Cf. A002293, A007226, A007228, A334640, A334643, A334682, A334785.

%K nonn,easy

%O 0,3

%A _Benjamin Hackl_, May 12 2020