A257387 Number of Motzkin paths of length n with no level steps at height 4.

1, 1, 2, 4, 9, 21, 51, 127, 323, 834, 2179, 5743, 15238, 40637, 108800, 292200, 786703, 2122387, 5735596, 15522682, 42064028, 114117541, 309918698, 842489130, 2292332265, 6242655886

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 21 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005043, A217312, A252354, A257386.

Programs

Mathematica

CoefficientList[Series[1/(1-x-x^2*(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 *)

PARI

x='x+O('x^50); Vec(1/(1-x-x^2*(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, Jun 03 2017

Formula

a(n) = a(n-1) + Sum_{j=0..n-2} A257386(j)*a(n-j).

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

a(n) ~ 3^(n+1/2)/(24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2015