OFFSET
0,2
LINKS
Stefano Spezia, Table of n, a(n) for n = 0..1000
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) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) for n > 0.
a(n) ~ c*2^(8*n)*3^(-3*n)/n^(3/2), where c = (11/9)*sqrt(2/(3*Pi)). - Stefano Spezia, Oct 19 2022
EXAMPLE
For n = 1, there are the 3_2-Dyck paths UDDD, DUDD, DDUD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.
MATHEMATICA
a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n+1) - Binomial[4*n+2, n]/(n+1); Array[a, 22, 0]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah Selkirk, May 11 2020
STATUS
approved