A257386 - OEIS

%I #20 Apr 09 2017 03:02:35

%S 1,1,2,4,9,21,51,126,316,799,2034,5202,13357,34407,88888,230237,

%T 597829,1555962,4058944,10612102,27807135,73025751,192204957,

%U 507025163,1340545113,3552492126

%N Number of Motzkin paths of length n with no level steps at height 3.

%H G. C. Greubel, <a href="/A257386/b257386.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = a(n-1) + Sum_{j=0..n-2} A252354(j)*a(n-j).

%F G.f: 1/(1-x-x^2*(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)/(50*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 24 2015

%t CoefficientList[Series[1/(1-x-x^2*(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 24 2015 *)

%o (PARI) x='x+O('x^50); Vec(1/(1-x-x^2*(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_, Apr 08 2017

%Y Cf. A005043, A217312, A252354.

%K nonn

%O 0,3

%A _José Luis Ramírez Ramírez_, Apr 21 2015