OFFSET
0,2
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.
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.
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
EXAMPLE
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).
MAPLE
b:= proc(x, y) option remember; `if`(x=y, x,
`if`(y+4<x, b(x-1, y+4), 0)+`if`(y>0, 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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrei Asinowski, May 08 2020
STATUS
approved