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%I #27 Feb 14 2017 22:11:30
%S 1,0,1,3,11,39,140,504,1823,6621,24144,88380,324699,1197045,4427565,
%T 16427385,61129025,228103185,853399640,3200710680,12032399045,
%U 45332769075,171148151095,647412581643,2453529142471,9314461044639,35419207688050,134894888442714,514506926871927
%N Number of 3-Motzkin paths of length n with no level steps at even level.
%H G. C. Greubel, <a href="/A257290/b257290.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.
%F G.f.: (1 - 3*z - sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2).
%F a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015
%F Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Sep 24 2016
%e For n=3 we have 3 paths: UH1D, UH2D, UH3D.
%t CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 21 2015 *)
%o (PARI) x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ _G. C. Greubel_, Feb 14 2017
%Y Cf. A090345, A025266.
%K nonn
%O 0,4
%A _José Luis Ramírez Ramírez_, Apr 20 2015
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