A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

0, 0, 9, 19, 72, 324, 1595, 8307, 44982, 250648, 1427679, 8274825, 48644310, 289334160, 1738043892, 10529070020, 64252519830, 394601627376, 2437058926871, 15126463230165, 94306717535940, 590318477063700, 3708527622652755, 23374587898663155, 147770791807427880

Offset

0,3

Comments

For n = 2, there is no 3rd up step, a(2) = 9 enumerates the total number of down steps between the 2nd up step and the end of the path.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1212

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Formula

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

Example

For n = 2, there are the 2-Dyck paths UUDDDD, UDUDDD, UDDUDD. Between the 2nd up step and the end of the path there are a(2) = 4 + 3 + 2 = 9 down steps in total.

Maple

b:= proc(x, y, u, c) option remember; `if`(x=0, c,
     `if`(y+2<x, b(x-1, y+2, min(u+1, 3), c), 0)+
     `if`(y>0, b(x-1, y-1, u, c+`if`(u=2, 1, 0)), 0))
    end:
a:= n-> b(3*n, 0$3):
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 9][n+1],
     (3*(n-1)*(3*n-8)*(3*n-7)*(13*n-20)*a(n-1))/
     (2*(13*n-33)*(n-2)*(2*n-3)*n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020

Mathematica

a[0] = a[1] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 2}]; Array[a, 25, 0] (* Amiram Eldar, May 09 2020 *)

Prog

(PARI) a(n) = if (n<=1, 0, 2*sum(j=1, 2, binomial(3*j+1, j) * binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

Crossrefs

Cf. A007226, A007228, A124724, A334641, A334642, A334682.

Sequence in context: A360429 A041158 A146080 * A259093 A186508 A000981

Adjacent sequences: A334637 A334638 A334639 * A334641 A334642 A334643

Keyword

nonn,easy

Author

Benjamin Hackl, May 07 2020

Status

approved