A334646 a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.

0, 0, 0, 118, 409, 2368, 15750, 112716, 845295, 6551208, 52035714, 421286280, 3463401007, 28832656408, 242565115858, 2058945519936, 17611312647075, 151647023490480, 1313460091978458, 11435310622320552, 100019000856225156, 878443730199290560

Offset

0,4

Comments

A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

Links

Table of n, a(n) for n=0..21.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Formula

a(0) = a(1) = a(2) = 0 and a(n) = 3*Sum_{j=1..3} binomial(4*j+1, j)*binomial(4*(n-j), n-j)/((4*j+1)*(n-j+1)) for n > 2.

Example

For n = 3, there is no 4th up step, a(3) = 118 enumerates the total number of down steps between the 3rd up step and the end of the path.

Mathematica

a[0] = a[1] = a[2] = 0; a[n_] := 3 * Sum[Binomial[4*j + 1, j] * Binomial[4*(n - j), n - j]/((4*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

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, 4)]) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

Crossrefs

Cf. A002293, A007226, A007228, A334641, A334645, A334682, A334785.

Sequence in context: A165462 A261464 A250793 * A232776 A175876 A220715

Adjacent sequences: A334643 A334644 A334645 * A334647 A334648 A334649

Keyword

nonn,easy

Author

Benjamin Hackl, May 12 2020

Status

approved