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A334645
a(n) is the total number of down steps between the 2nd and 3rd up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.
5
0, 0, 18, 52, 277, 1752, 12120, 88692, 674751, 5282160, 42267384, 344152080, 2842055359, 23746693240, 200383750632, 1705243729560, 14617677294675, 126106202849760, 1094034474058488, 9538676631305712, 83536778390997780, 734521734171474400, 6481894477750488160
OFFSET
0,3
COMMENTS
For n = 2, there is no 3rd up step, a(2) = 18 enumerates the total number of down steps between the 2nd up step and the end of the path.
LINKS
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
FORMULA
a(0) = a(1) = 0 and a(n) = 3*Sum_{j=0..2} binomial(4*j+1, j) * binomial(4*(n-j), n-j)/((4*j+1) * (n-j+1)) for n > 1.
EXAMPLE
For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 4 + 5 + 6 = 18 down steps between the 2nd up step and the end of the path.
PROG
(SageMath) [3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 3)]) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020
KEYWORD
nonn,easy
AUTHOR
Benjamin Hackl, May 12 2020
STATUS
approved