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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 26 2003
STATUS
approved