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%I #11 Jun 03 2017 19:01:42
%S 1,1,2,4,9,21,51,127,322,827,2145,5607,14751,39020,103713,276848,
%T 741901,1995340,5384554,14576673,39579527,107776557,294283193,
%U 805649528,2211176173,6083560542,16776970140,46372110274,128456563024,356600559820,991986172469,2765030171165,7722156349298,21607098380159
%N Number of Motzkin paths of length n with no peaks at level 4.
%H G. C. Greubel, <a href="/A257519/b257519.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1/(1-x-x^2/(1-x-x^2/(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)/(98*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 27 2015
%e For n=4 we have 9 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH, HHUD and UUDD
%t CoefficientList[Series[1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)))))), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Apr 27 2015 *)
%o (PARI) x='x+O('x^50); Vec(1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))))))) \\ _G. C. Greubel_, Jun 03 2017
%Y Cf. A089372, A257300, A257104.
%K nonn
%O 0,3
%A _José Luis Ramírez Ramírez_, Apr 27 2015
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