A257517 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at even level.

1, 0, 1, 0, 2, 2, 5, 8, 18, 30, 66, 120, 252, 484, 1005, 1984, 4110, 8278, 17150, 35024, 72748, 150012, 312642, 649424, 1358244, 2837484, 5954980, 12497616, 26313432, 55434248, 117062205, 247412928, 523881238, 1110335334, 2356819254, 5007428384, 10652412108, 22682131308, 48349084054, 103150869360, 220276819836

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 27 2015

Keywords

• nonn
• easy

Crossrefs

Cf. A257516, A005572, A025266.

Programs

Mathematica

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 *)

Formula

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

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