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%I #33 Dec 22 2021 02:54:48
%S 1,1,2,4,9,22,56,147,393,1065,2915,8042,22330,62339,174837,492313,
%T 1391134,3943130,11207594,31934552,91197474,260969372,748176873,
%U 2148622932,6180146228,17801978083,51347929943,148293450023,428774359142,1241110916678
%N G.f.: (1-x+sqrt(1-2*x-3*x^2))/(1-3*x+x^2+x^3+(1-x^2)*sqrt(1-2*x-3*x^2)).
%H G. C. Greubel, <a href="/A244886/b244886.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H J.-L. Baril and A. Petrossian, <a href="https://doi.org/10.1016/j.disc.2014.12.003">Equivalence classes of Dyck paths modulo some statistics</a>, Discrete Mathematics, Volume 338, Issue 4, 6 April 2015, Pages 655-660. See Theorem 1.
%H Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, <a href="http://jl.baril.u-bourgogne.fr/barasi2.pdf">Equivalence Classes of Skew Dyck Paths Modulo some Patterns</a>, 2021.
%H K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://arxiv.org/abs/1510.01952">Equivalence classes of ballot paths modulo strings of length 2 and 3</a>, arXiv:1510.01952 [math.CO], 2015.
%F a(n) ~ 3^(n+7/2)/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 10 2014
%F Conjecture D-finite with recurrence: n*a(n) +(-5*n+3)*a(n-1) +2*(n)*a(n-2) +(13*n-30)*a(n-3) +3*(-1)*a(n-4) +(-8*n+21)*a(n-5) +3*(-n+3)*a(n-6)=0. - _R. J. Mathar_, Jan 24 2020
%t CoefficientList[Series[(1 - x + Sqrt[1 - 2 x - 3 x^2])/(1 - 3 x + x^2 + x^3 + (1 - x^2) Sqrt[1 - 2 x - 3 x^2]), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 10 2014 *)
%o (PARI) my(x='x+O('x^50)); Vec((1-x+sqrt(1-2*x-3*x^2))/(1-3*x+x^2+x^3+(1-x^2)*sqrt(1-2*x-3*x^2))) \\ _G. C. Greubel_, Apr 05 2017
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jul 09 2014
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