

A334786


a(n) is the total number of down steps before the first up step in all 4_2Dyck paths of length 5*n. A 4_2Dyck path is a lattice path with steps (1, 4), (1, 1) that starts and ends at y = 0 and stays above the line y = 2.


4



0, 3, 16, 115, 950, 8510, 80388, 788392, 7950930, 81935425, 859005840, 9132977490, 98240702586, 1067197649840, 11691092372000, 129011823098160, 1432744619523530, 16000911127589355, 179590878292003200, 2024687100104286525, 22917687021180660940
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OFFSET

0,2


LINKS

Table of n, a(n) for n=0..20.
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.


EXAMPLE

For n = 1, there are the 4_2Dyck 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

Cf. A001764, A002293, A002294, A334785, A334787.
Sequence in context: A329113 A042437 A324514 * A159606 A211210 A177402
Adjacent sequences: A334783 A334784 A334785 * A334787 A334788 A334789


KEYWORD

nonn,easy


AUTHOR

Sarah Selkirk, May 11 2020


STATUS

approved



