OFFSET
0,2
COMMENTS
A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.
For n = 1, there is no 2nd up step, a(1) = 5 enumerates the total number of down steps between the 1st up step and the end of the path.
LINKS
Andrei Asinowski, Benjamin Hackl, Sarah J. 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) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.
EXAMPLE
For n = 1, the 3_1-Dyck paths are UDDD, DUDD. This corresponds to a(1) = 3 + 2 = 5 down steps between the 1st up step and the end of the path.
For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 = 16 down steps between the 1st and 2nd up step.
MATHEMATICA
a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - 2 * Binomial[4*n + 1, n]/(n + 1) + 6 * Binomial[4*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)
PROG
(SageMath) [3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benjamin Hackl, May 12 2020
STATUS
approved