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 A334641 #23 Aug 08 2020 01:57:09
%S A334641 0,0,0,43,108,444,2099,10683,56994,314296,1776519,10236081,59892690,
%T A334641 354886920,2125117332,12839859620,78176677734,479177993904,
%U A334641 2954360065247,18309779343549,114001476318240,712751759478780,4472908385838795,28165267333869435
%N A334641 a(n) is the total number of down steps between the 3rd and 4th up steps in all 2-Dyck paths of length 3*n.
%C A334641 A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.
%C A334641 For n = 3, there is no 4th up step, a(3) = 43 enumerates the total number of down steps between the 3rd up step and the end of the path.
%H A334641 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 A334641 a(0) = a(1) = a(2) = 0 and a(n) = 2*Sum_{j=1..3}binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)) for n > 2.
%t A334641 a[0] = a[1] = a[2] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 24, 0] (* _Amiram Eldar_, May 09 2020 *)
%o A334641 (PARI) a(n) = if (n<=2, 0, 2*sum(j=1, 3, binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ _Michel Marcus_, May 09 2020
%Y A334641 Cf. A007226, A007228, A124724, A334640, A334642, A334682.
%K A334641 nonn,easy
%O A334641 0,4
%A A334641 _Benjamin Hackl_, May 07 2020