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 A334610 #22 Apr 28 2023 12:18:38 %S A334610 0,7,58,505,4650,44677,443238,4507461,46744100,492492330,5257084420, %T A334610 56734340091,618001356458,6785943435960,75033214770640, %U A334610 834733624099485,9336542892778440,104932793226255165,1184421713336050590,13421053387405062290,152613573227667516580 %N A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5*n (n up-steps and 4*n down-steps). %C A334610 A 4_1-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -1, and ends at the x-axis. %H A334610 Stefano Spezia, <a href="/A334610/b334610.txt">Table of n, a(n) for n = 0..900</a> %H A334610 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 A334610 a(n) = 2*binomial(5*(n+1)+2, n+1)/(5*(n+1)+2) - 4*binomial(5*n+2, n)/(5*n+2). %F A334610 G.f.: ((1 - 2*x)*HypergeometricPFQ([2/5, 3/5, 4/5, 6/5], [3/4, 5/4, 3/2], 3125*x/256) - 1)/x. - _Stefano Spezia_, Apr 25 2023 %e A334610 For n=1, a(1) = 7 is the total number of down-steps after the last up-step in Udddd, dUddd. %t A334610 a[n_] := 2 * Binomial[5*n + 7, n + 1]/(5*n + 7) - 4 * Binomial[5*n + 2, n]/(5*n + 2); Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *) %Y A334610 Cf. A002294, A334651, A334719, A334611, A334612. %K A334610 nonn %O A334610 0,2 %A A334610 _Andrei Asinowski_, May 13 2020