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 A089354 Number of generalized {(1,2),(1,-1)}-Dyck paths of length 3n with no peaks at level 2. 1
 1, 0, 1, 4, 19, 96, 508, 2780, 15607, 89392, 520337, 3069232, 18305876, 110214144, 668950744, 4088824140, 25146253311, 155491812384, 966142729939, 6029139839684, 37771401328459, 237467581184384, 1497754198565104 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified September 10 08:45 EDT 2024. Contains 375786 sequences. (Running on oeis4.)