OFFSET
0,3
COMMENTS
For n = 1, there is no (n-1)-th up step, a(1) = 1 is the total number of down steps before the first up step.
LINKS
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) = 2*binomial(3*n+5, n+1)/(3*n+5) - 6*binomial(3*n+2, n)/(3*n+2) for n > 0.
EXAMPLE
For n = 2, the 2_1-Dyck paths are UDDDUD, UDDUDD, UDUDDD, UUDDDD, DUDDUD, DUDUDD, DUUDDD. Therefore the total number of down steps between the first and second up step is a(2) = 3 + 2 + 1 + 0 + 2 + 1 +0 = 9.
MATHEMATICA
a[0] = 0; a[n_] := 2*Binomial[3*n+5, n+1]/(3*n + 5) - 6 * Binomial[3*n + 2, n]/(3*n + 2); Array[a, 24, 0]
PROG
(SageMath) [2*binomial(3*n + 5, n + 1)/(3*n + 5) - 6*binomial(3*n + 2, n)/(3*n + 2) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah Selkirk, May 18 2020
STATUS
approved