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%I #40 Sep 20 2021 00:58:03
%S 1,1,1,2,5,12,30,76,196,512,1353,3610,9713,26324,71799,196938,542895,
%T 1503312,4179603,11662902,32652735,91695540,258215664,728997192,
%U 2062967382,5850674704,16626415975,47337954326,135015505407,385719506620,1103642686382
%N Expansion of (-2 +x^2 +x -x*sqrt(1-2*x-3*x^2))/(-1 +x -sqrt(1-2*x-3*x^2)).
%C For n > 1, a(n) is the number of Motzkin n-paths that start with an up step. - _Gennady Eremin_, Sep 18 2021
%H G. C. Greubel, <a href="/A244884/b244884.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="http://jl.baril.u-bourgogne.fr/equivdyck.pdf">Equivalence classes of Dyck paths modulo some statistics</a>, 2014.
%H Gennady Eremin, <a href="https://arxiv.org/abs/2002.08067">Generating function for Naturalized Series: The case of Ordered Motzkin Words</a>, arXiv:2002.08067 [math.CO], 2020.
%F a(n) ~ 3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jul 10 2014
%F Conjecture D-finite with recurrence: (n+2)*a(n) +(-3*n-1)*a(n-1) -n*a(n-2) +3*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jan 24 2020
%F G.f.: x + (1-x)*M(x), where M(x) is the g.f. of A001006. - _Gennady Eremin_, Feb 14 2021
%t CoefficientList[Series[(-2 + x^2 + x - x Sqrt[1 - 2 x - 3 x^2])/(-1 + x - Sqrt[1 - 2 x - 3 x^2]), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 10 2014 *)
%o (PARI) my(x='x + O('x^50)); Vec((-2 +x^2 +x -x*sqrt(1-2*x-3*x^2))/(-1 +x -sqrt(1-2*x-3*x^2))) \\ _G. C. Greubel_, Feb 14 2017
%Y Apart from initial terms, same as A002026 and A105695.
%Y Cf. A001006.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Jul 09 2014