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%I #29 Feb 14 2017 17:47:32
%S 1,1,2,4,9,20,46,106,248,584,1389,3329,8047,19607,48167,119287,297829,
%T 749632,1902044,4864553,12538933,32568528,85224251,224618900,
%U 596106393,1592429464,4280667705,11575188106,31474407317,86029586086,236292044931,651952466845
%N Number of Motzkin paths of length n with no level steps at height 2.
%H G. C. Greubel, <a href="/A252354/b252354.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = a(n-1) + Sum_{j=0..n-2} A217312(j)*a(n-j).
%F G.f: 1/(1-x-x^2(1/(1-x-x^2*R(x)))), where R(x) is the g.f. of Riordan numbers (A005043).
%F a(n) ~ 3^(n+3/2) / (32*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015
%F Conjecture: (-n+3)*a(n) +3*(2*n-7)*a(n-1) +(-7*n+24)*a(n-2) +2*(-7*n+36)*a(n-3) +2*(11*n-51)*a(n-4) +3*(3*n-23)*a(n-5) +(-10*n+63)*a(n-6) +3*(n-6)*a(n-7)=0. - _R. J. Mathar_, Sep 24 2016
%t CoefficientList[Series[1/(1-x-x^2(1/(1-x-x^2*(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x))))), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 21 2015 *)
%o (PARI) x='x + O('x^50); Vec(1/(1-x-x^2*(1/(1-x-x^2*(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)))))) \\ _G. C. Greubel_, Feb 14 2017
%Y Cf. A005043, A217312.
%K nonn
%O 0,3
%A _José Luis Ramírez Ramírez_, Apr 20 2015
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