A334612 - OEIS

%I #21 Apr 28 2023 11:22:13

%S 0,10,100,955,9296,92704,944636,9801929,103262436,1101802764,

%T 11883775540,129365990061,1419569592748,15686292728288,

%U 174399501150236,1949516926153045,21898270953801720,247045453792464294,2797968888077323968,31801559116255638374,362622937212800684560

%N a(n) is the total number of down-steps after the final up-step in all 4_3-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

%C A 4_3-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -3, and ends at the x-axis.

%H Stefano Spezia, <a href="/A334612/b334612.txt">Table of n, a(n) for n = 0..900</a>

%H 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 a(n) = 4*binomial(5*(n+1)+4, n+1)/(5*(n+1)+4) - 16*binomial(5*n+4, n)/(5*n+4).

%F G.f.: (4 - 21*x - 4*(1 - 4*x)*HypergeometricPFQ([-1/5, 1/5, 2/5, 3/5], [1/4, 1/2, 3/4], 3125*x/256))/(5*x^2). - _Stefano Spezia_, Apr 25 2023

%e For n = 1, a(1) = 10 is the total number of down-steps after the last up-step in Udddd, dUddd, ddUdd, dddUd.

%t a[n_] := 4 * Binomial[5*n + 9, n + 1]/(5*n + 9) - 16 * Binomial[5*n + 4, n]/(5*n + 4); Array[a, 21, 0] (* _Amiram Eldar_, May 13 2020 *)

%Y Cf. A334787, A334719, A334610, A334611.

%K nonn

%O 0,2

%A _Andrei Asinowski_, May 13 2020