%I #11 Mar 06 2022 08:13:26

%S 1,0,1,0,2,2,5,8,18,30,66,120,252,484,1005,1984,4110,8278,17150,35024,

%T 72748,150012,312642,649424,1358244,2837484,5954980,12497616,26313432,

%U 55434248,117062205,247412928,523881238,1110335334,2356819254,5007428384,10652412108,22682131308,48349084054,103150869360,220276819836

%N Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at even level.

%F G.f.: (1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3)))/(2*x^2).

%F D-finite with recurrence +(n+2)*(n^2-n+3)*a(n) +(n+1)*(n^2+1)*a(n-1) -4*(n-1)*(n^2-n+3)*a(n-2) +2*(-4*n^3+11*n^2-13*n+19)*a(n-3) -2*(2*n-7)*(n^2+1)*a(n-4) +4*(2*n-11)*(n^2-n+3)*a(n-5) +4*(3*n^3-21*n^2+12*n-34)*a(n-6) +4*(n-8)*(n^2+1)*a(n-7)=0. - _R. J. Mathar_, Jun 07 2016

%t CoefficientList[Series[(1-2*x^3-Sqrt[(1-2*x^3)*(1-4*x^2-2*x^3)])/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 27 2015 *)

%Y Cf. A257516, A005572, A025266.

%K nonn,easy

%O 0,5

%A _José Luis Ramírez Ramírez_, Apr 27 2015