A089354 Number of generalized {(1,2),(1,-1)}-Dyck paths of length 3n with no peaks at level 2.

1, 0, 1, 4, 19, 96, 508, 2780, 15607, 89392, 520337, 3069232, 18305876, 110214144, 668950744, 4088824140, 25146253311, 155491812384, 966142729939, 6029139839684, 37771401328459, 237467581184384, 1497754198565104, 9474388380156944, 60093935844627364

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Formula

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

G.f.: g/(1+zg^2), where g=1+zg^3, g(0)=1. Also g=2*sin(arcsin(3*sqrt(3z)/2)/3)/sqrt(3z).

a(n) ~ 3^(3*n+3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n+5)). - Vaclav Kotesovec, Mar 17 2014

Conjecture D-finite with recurrence 64*n*(2*n+1)*a(n) +8*(-142*n^2+205*n-66)*a(n-1) +2*(880*n^2-3901*n+3924)*a(n-2) +57*(3*n-5)*(3*n-7)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

a(n) = Sum_{k=0..n} (-1)^k * (2*k+1) * binomial(3*n-k+1,n-k)/(3*n-k+1). - Seiichi Manyama, Nov 08 2025

From Seiichi Manyama, Nov 29 2025: (Start)

G.f.: 1/(1 - x^2*g^4), where g = 1+x*g^3 is the g.f. of A001764.

a(0) = 1; a(n) = 4 * Sum_{k=0..floor(n/2)} k * binomial(3*n-2*k,n-2*k)/(3*n-2*k).

a(0) = 1; a(n) = (2/n) * Sum_{k=0..floor(n/2)} k * binomial(3*n-2*k-1,n-2*k). (End)

Example

a(3)=4 because we have UUDUDDDDD, UUUDDDDDD, UUDDUDDDD and UUDDDUDDD, where U=(1,2) and D=(1,-1).

Mathematica

Flatten[{1, Table[2/n*Sum[(-2)^i*(i+1)*Binomial[3*n+1, n-2-i], {i, 0, n-2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 17 2014 *)

Crossrefs

Cf. A000957, A001764, A390500, A390744.

Sequence in context: A020060 A122394 A047781 * A217217 A083315 A301417

Adjacent sequences: A089351 A089352 A089353 * A089355 A089356 A089357

Keyword

nonn

Author

Emeric Deutsch, Dec 26 2003

Status

approved