OFFSET
0,2
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.
FORMULA
a(0) = 0 and a(n) = 4 * binomial(5*n, n)/(n+1) - 2 * binomial(5*n+2, n)/(n+1) for n > 0.
a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (7/16)*sqrt(5/(2*Pi)). - Stefano Spezia, Oct 19 2022
EXAMPLE
For n = 1, there are the 4_2-Dyck paths UDDDD, DUDDD, DDUDD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.
MATHEMATICA
a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - 2 * Binomial[5*n+2, n]/(n+1); Array[a, 21, 0]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah Selkirk, May 11 2020
STATUS
approved