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 A334608 #19 Aug 06 2020 10:01:15 %S A334608 0,5,34,236,1714,12922,100300,796572,6443536,52909593,439896626, %T A334608 3695917940,31331587252,267669458420,2302188456120,19918434257052, %U A334608 173240112503520,1513821095788420,13283883136738344,117009704490121520,1034217260142108570,9169842145476773250,81537271617856588380 %N A334608 a(n) is the total number of down-steps after the final up-step in all 3_1-Dyck paths of length 4*n (n up-steps and 3n down-steps). %C A334608 A 3_1-Dyck path is a lattice path with steps U=(1, 3), d=(1, -1) that starts at (0,0), stays (weakly) above y=-1, and ends at the x-axis. %H A334608 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 A334608 a(n) = 2*binomial(4*(n+1)+2, n+1)/(4*(n+1)+2) - 4*binomial(4*n+2, n)/(4*n+2). %e A334608 For n=1, a(1)=5 is the total number of down-steps after the last up-step in Uddd, dUdd. %t A334608 a[n_] := 2 * Binomial[4*n + 6, n + 1]/(4*n + 6) - 4 * Binomial[4*n + 2, n]/(4*n + 2); Array[a, 23, 0] (* _Amiram Eldar_, May 13 2020 *) %o A334608 (SageMath) [2*binomial(4*(n + 1) + 2, n + 1)/(4*(n + 1) + 2) - 4*binomial(4*n + 2, n)/(4*n + 2) for n in srange(30)] # _Benjamin Hackl_, May 13 2020 %Y A334608 Cf. A002293, A007226, A007228, A334609, A334645, A334646, A334647, A334648, A334649, A334680, A334682, A334785. %K A334608 nonn,easy %O A334608 0,2 %A A334608 _Andrei Asinowski_, May 13 2020