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A257363 Number of 3-Motzkin paths with no level steps at height 1.

%I A257363 #25 Jun 02 2025 11:47:02
%S A257363 1,3,10,33,110,369,1247,4248,14603,50724,178314,635526,2300829,
%T A257363 8477382,31842897,122103276,478372886,1915188093,7831613468,
%U A257363 32674683984,138871668314,600140517762,2631926843602,11690520554421,52498671870181,237966449687118,1087246253873875,5001141997115010,23137102115963262
%N A257363 Number of 3-Motzkin paths with no level steps at height 1.
%C A257363 For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(2)H(1), H(2)H(2), H(1)H(3), H(3)H(1), H(3)H(3), H(2)H(3), H(3)H(2), UD.
%H A257363 Robert Israel, <a href="/A257363/b257363.txt">Table of n, a(n) for n = 0..1000</a>
%F A257363 G.f.: 1/(1-3*x-x*F(x)), where F(x) is the g.f. of the sequence A117641.
%F A257363 G.f.: 2*(3+x)/(6-17*x-9*x^2+x*sqrt(1-6*x+5*x^2)).
%F A257363 a(n) ~ 5^(n+3/2)/(98*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 21 2015
%F A257363 From _Robert Israel_, Apr 28 2015 (Start):
%F A257363 G.f.: (6-x*sqrt(1-6*x+5*x^2)-17*x-9*x^2)/(6-36*x+42*x^2+38*x^3).
%F A257363 3*(-n+1)*a(n) +9*(4*n-7)*a(n-1) +9*(-16*n+39)*a(n-2) +(197*n-656)*a(n-3) +9*(n+15)*a(n-4) +95*(-n+4)*a(n-5)=0. (End)
%p A257363 rec:= (95+95*n)*a(n)+(-180-9*n)*a(n+1)+(-329-197*n)*a(n+2)+(369+144*n)*a(n+3)+(-117-36*n)*a(4+n)+(12+3*n)*a(n+5):
%p A257363 f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=10,a(3)=33,a(4)=110},a(n),remember):
%p A257363 seq(f(n),n=0..100); # _Robert Israel_, Apr 28 2015
%t A257363 CoefficientList[Series[2*(3+x)/(6-17*x-9*x^2+x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 21 2015 *)
%Y A257363 Cf. A117641, A217312, A253831, A059231.
%K A257363 nonn
%O A257363 0,2
%A A257363 _José Luis Ramírez Ramírez_, Apr 20 2015