A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

0, 0, 34, 132, 722, 4638, 32416, 238956, 1827918, 14370595, 115384756, 942115942, 7798224226, 65286060253, 551838621972, 4702955036640, 40366238473530, 348631520142879, 3027590307082804, 26420699531880832, 231571468023697960, 2037650653547067005

Offset

0,3

Comments

A 3_1-Dyck 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 = 2, there is no 3rd up step, a(2) = 34 enumerates the total number of down steps between the 2nd 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, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Formula

a(0) = a(1) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..2} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 9*[n=2] for n > 1, where [ ] is the Iverson bracket.

Example

For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 34 down steps between the 2nd up step and the end of the path.

Mathematica

a[0] = a[1] = 0; a[n_] := 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)), {j, 1, 2}] - 9 * Boole[n == 2]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

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, 3)]) - 9*(n==2) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

Crossrefs

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334649, A334680, A334682, A334785.

Sequence in context: A293039 A216308 A010021 * A044366 A044747 A172001

Adjacent sequences: A334645 A334646 A334647 * A334649 A334650 A334651

Keyword

nonn,easy

Author

Benjamin Hackl, May 12 2020

Status

approved