A334612 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).

0, 10, 100, 955, 9296, 92704, 944636, 9801929, 103262436, 1101802764, 11883775540, 129365990061, 1419569592748, 15686292728288, 174399501150236, 1949516926153045, 21898270953801720, 247045453792464294, 2797968888077323968, 31801559116255638374, 362622937212800684560

Offset: 0

Author

Andrei Asinowski, May 13 2020

Keywords

• nonn

Comments

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.

Examples

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

Links

• Stefano Spezia, Table of n, a(n) for n = 0..900
• A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Crossrefs

Cf. A334787, A334719, A334610, A334611.

Programs

Mathematica

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 *)

Formula

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

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