

A334649


a(n) is the total number of down steps between the third and fourth up steps in all 3_1Dyck paths of length 4*n.


4



0, 0, 0, 236, 1034, 6094, 40996, 295740, 2231022, 17370163, 138473536, 1124433142, 9266859394, 77307427741, 651540030688, 5538977450256, 47442103851930, 409000732566399, 3546232676711824, 30903652601552272, 270529448396053576, 2377829916885541565
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OFFSET

0,4


COMMENTS

A 3_1Dyck 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 = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path.


LINKS

Table of n, a(n) for n=0..21.
Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Downstep statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.


FORMULA

a(0) = a(1) = a(2) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(nj), nj)/((4*j+2)*(nj+1))  52*[n=3] for n > 2, where [ ] is the Iverson bracket.


PROG

(SageMath) [binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n  j), n  j)/(4*j + 2)/(n  j + 1) for j in srange(1, 4)])  52*(n==3) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020


CROSSREFS

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785.
Sequence in context: A138819 A273181 A259639 * A233983 A251491 A251484
Adjacent sequences: A334646 A334647 A334648 * A334650 A334651 A334652


KEYWORD

nonn,easy


AUTHOR

Benjamin Hackl, May 12 2020


STATUS

approved



