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%I #21 Sep 15 2020 05:53:27
%S 1,0,1,4,19,96,508,2780,15607,89392,520337,3069232,18305876,110214144,
%T 668950744,4088824140,25146253311,155491812384,966142729939,
%U 6029139839684,37771401328459,237467581184384,1497754198565104
%N Number of generalized {(1,2),(1,-1)}-Dyck paths of length 3n with no peaks at level 2.
%H G. C. Greubel, <a href="/A089354/b089354.txt">Table of n, a(n) for n = 0..1000</a>
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F a(n) = (2/n)*Sum_{i=0..(n-2)} (-2)^i*(i+1)*binomial(3n+1, n-2-i), n >= 1.
%F 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).
%F a(n) ~ 3^(3*n+3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n+5)). - _Vaclav Kotesovec_, Mar 17 2014
%F 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
%e a(3)=4 because we have UUDUDDDDD, UUUDDDDDD, UUDDUDDDD and UUDDDUDDD, where U=(1,2) and D=(1,-1).
%t 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 *)
%K nonn
%O 0,4
%A _Emeric Deutsch_, Dec 26 2003
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