

A334977


a(n) is the total number of down steps between the (n1)th and nth up steps in all 2_1Dyck paths of length 3*n. A 2_1Dyck path is a lattice path with steps (1, 2), (1, 1) that starts and ends at y = 0 and stays above the line y = 1.


4



0, 1, 9, 53, 299, 1692, 9690, 56221, 330165, 1959945, 11745435, 70974252, 432019844, 2646716264, 16307880462, 100996570221, 628356589721, 3925544432355, 24616047166095, 154886752443885, 977595783524955, 6187863825170160, 39269844955755960, 249819662230403148
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OFFSET

0,3


COMMENTS

For n = 1, there is no (n1)th up step, a(1) = 1 is the total number of down steps before the first up step.


LINKS

Table of n, a(n) for n=0..23.
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_1Dyck 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

Cf. A334976, A334978, A334979, A334980.
Sequence in context: A122588 A277999 A295203 * A038761 A003698 A001688
Adjacent sequences: A334974 A334975 A334976 * A334978 A334979 A334980


KEYWORD

nonn,easy


AUTHOR

Sarah Selkirk, May 18 2020


STATUS

approved



