

A334651


a(n) is the total number of down steps between the first and second up steps in all 4_1Dyck paths of length 5*n.


1



0, 7, 25, 155, 1195, 10282, 94591, 910480, 9054965, 92310075, 959473878, 10129715890, 108327387675, 1170975480360, 12773887368040, 140445927510832, 1554748206904325, 17314584431331025, 193849445090545875, 2180550929942519685, 24632294533221865028
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OFFSET

0,2


COMMENTS

A 4_1Dyck path is a lattice path with steps (1, 4), (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) = 7 enumerates the total number of down steps between the 1st up step and the end of the path.


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)  3*binomial(5*n+1, n)/(n+1) + 8*binomial(5*(n1), n1)/n  2*[n=1] for n > 0, where [ ] is the Iverson bracket.


EXAMPLE

For n = 1, the 4_1Dyck paths are DUDDD, UDDDD. This corresponds to a(1) = 3 + 4 = 7 down steps between the 1st up step and the end of the path.


MATHEMATICA

a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n + 1)  3 * Binomial[5*n + 1, n]/(n + 1) + 8*Binomial[5*(n  1), n  1]/n  2 * Boole[n == 1]; Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)


PROG

(SageMath) [4*binomial(5*n, n)/(n + 1)  3*binomial(5*n + 1, n)/(n + 1) + 8*binomial(5*(n  1), n  1)/n  2*(n==1) if n > 0 else 0 for n in srange(30)]


CROSSREFS

Cf. A002294, A124724, A334642, A334647, A334719, A334786, A334787.
Sequence in context: A241714 A151491 A208425 * A191237 A088009 A293532
Adjacent sequences: A334648 A334649 A334650 * A334652 A334653 A334654


KEYWORD

nonn,easy


AUTHOR

Benjamin Hackl, May 13 2020


STATUS

approved



