This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334643 #13 Aug 07 2020 12:07:56
%S A334643 0,0,16,53,209,963,4816,25367,138531,777041,4449511,25901655,
%T A334643 152818458,911755012,5491420104,33343242196,203881825163,
%U A334643 1254342228285,7759025239189,48227078649155,301056318504165,1886647802277315,11864793375611820,74854437302309175
%N A334643 a(n) is the total number of down steps between the second and third up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.
%C A334643 For n = 2, there is no 3rd up step, a(2) = 16 enumerates the total number of down steps between the 2nd up step and the end of the path.
%H A334643 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 A334643 a(0) = a(1) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..2}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 7*[n=2] for n > 1, where [ ] is the Iverson bracket.
%e A334643 For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 4 + 3 + 2 + 1 + 1 + 2 + 3 = 16 down steps between the 2nd up step and the end of the path.
%o A334643 (SageMath) [binomial(3*n + 1, n)/(3*n + 1) + 4*sum([binomial(3*j + 2, j)*binomial(3*(n - j), n - j)/(3*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 7*(n==2) if n >= 2 else 0 for n in srange(30)] # _Benjamin Hackl_, May 12 2020
%Y A334643 Cf. A001764, A007226, A030983, A334640, A334642, A334644.
%K A334643 nonn,easy
%O A334643 0,3
%A A334643 _Benjamin Hackl_, May 12 2020