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 A334651 #14 Aug 07 2020 12:08:06 %S A334651 0,7,25,155,1195,10282,94591,910480,9054965,92310075,959473878, %T A334651 10129715890,108327387675,1170975480360,12773887368040, %U A334651 140445927510832,1554748206904325,17314584431331025,193849445090545875,2180550929942519685,24632294533221865028 %N A334651 a(n) is the total number of down steps between the first and second up steps in all 4_1-Dyck paths of length 5*n. %C A334651 A 4_1-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. %C A334651 For n = 1, there is no 2nd up step, a(1) = 7 enumerates the total number of down steps between the 1st up step and the end of the path. %H A334651 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 A334651 a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - 3*binomial(5*n+1, n)/(n+1) + 8*binomial(5*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket. %e A334651 For n = 1, the 4_1-Dyck paths are DUDDD, UDDDD. This corresponds to a(1) = 3 + 4 = 7 down steps between the 1st up step and the end of the path. %t A334651 a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n + 1) - 3 * Binomial[5*n + 1, n]/(n + 1) + 8*Binomial[5*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *) %o A334651 (SageMath) [4*binomial(5*n, n)/(n + 1) - 3*binomial(5*n + 1, n)/(n + 1) + 8*binomial(5*(n - 1), n - 1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] %Y A334651 Cf. A002294, A124724, A334642, A334647, A334719, A334786, A334787. %K A334651 nonn,easy %O A334651 0,2 %A A334651 _Benjamin Hackl_, May 13 2020