A334611 a(n) is the total number of down-steps after the final up-step in all 4_2-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

0, 9, 82, 747, 7065, 69098, 694272, 7127865, 74468546, 789265125, 8466019380, 91736269053, 1002710879409, 11042713886256, 122413333216960, 1364880618458565, 15296452128008100, 172218124701600741, 1946960139291303222, 22092883135853433030, 251545025683283255770

Offset: 0

Author

Andrei Asinowski, May 13 2020

Keywords

• nonn

Comments

A 4_2-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.

Examples

			For n=1, a(1) = 9 is the total number of down-steps after the last up-step in Udddd, dUddd, ddUdd.
		

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. A334786, A334719, A334610, A334612.

Programs

Mathematica

a[n_] := 3 * Binomial[5*n + 8, n + 1]/(5*n + 8) - 9 * Binomial[5*n + 3, n]/(5*n + 3); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

Formula

a(n) = 3*binomial(5*(n+1)+3, n+1)/(5*(n+1)+3) - 9*binomial(5*n+3, n)/(5*n+3).

G.f.: ((1 - 3*x)*HypergeometricPFQ([3/5, 4/5, 6/5, 7/5], [5/4, 3/2, 7/4], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023