%I #31 Feb 14 2017 18:08:01
%S 1,1,2,4,8,17,38,88,210,514,1285,3270,8447,22100,58455,156077,420153,
%T 1139155,3108095,8527675,23514124,65127571,181111940,505487115,
%U 1415502195,3975790024,11197966459,31619946886,89496047586,253858251337,721531869889,2054639741185
%N Number of Motzkin paths of length n with no peaks at level 2.
%H G. C. Greubel, <a href="/A257300/b257300.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1/(1-x-x^2/(1-x+x^2*(1-M(x)))), where M(x) is the g.f. of Motzkin numbers A001006.
%F a(n) ~ 3^(n+7/2) / (50*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015
%F Conjecture: n*a(n) +(-5*n+3)*a(n-1) +6*(n)*a(n-2) +2*(n-9)*a(n-3) +6*(-n+4)*a(n-4) +(n-6)*a(n-5) +3*(-n+3)*a(n-6)=0. - _R. J. Mathar_, Sep 24 2016
%e For n=4 we have 8 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH and HHUD.
%t CoefficientList[Series[1/(1-x-x^2/(1-x+x^2*(1-(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)))), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 21 2015 *)
%o (PARI) x='x + O('x^50); Vec(1/(1-x-x^2/(1-x+x^2*(1-(1-x-sqrt(1-2*x-3*x^2))/(2*x^2))))) \\ _G. C. Greubel_, Feb 14 2017
%Y Cf. A089372.
%K nonn
%O 0,3
%A _José Luis Ramírez Ramírez_, Apr 20 2015