A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

0, 4, 30, 250, 2245, 21221, 208129, 2098565, 21619910, 226593015, 2408424760, 25899375645, 281273231985, 3080585212120, 33986840371400, 377364606387005, 4213620859310140, 47284625533425750, 532996618440511710, 6032169040263819485, 68517222947120776290

Offset: 0

Author

Andrei Asinowski, May 08 2020

Keywords

• nonn
• easy

Comments

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

Examples

			For n = 2, the a(2) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).
		

Links

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

Crossrefs

First order differences of A002294. Cf. A062985.

Cf. A334682 (similar for 3-Dyck paths).

Programs

Maple

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+40, b(x-1, y-1), 0))
    end:
a:= n-> b(5*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
      (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
       a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
       (2869*n^3-3253*n^2+1138*n-120)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

Mathematica

a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

PARI

a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ Andrew Howroyd, May 08 2020

SageMath

[binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # Benjamin Hackl, May 13 2020

Formula

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

a(n) = A062985(n+1, 4*n-1).

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