A252354 Number of Motzkin paths of length n with no level steps at height 2.

1, 1, 2, 4, 9, 20, 46, 106, 248, 584, 1389, 3329, 8047, 19607, 48167, 119287, 297829, 749632, 1902044, 4864553, 12538933, 32568528, 85224251, 224618900, 596106393, 1592429464, 4280667705, 11575188106, 31474407317, 86029586086, 236292044931, 651952466845

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

• nonn

Links

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

Crossrefs

Cf. A005043, A217312.

Programs

Mathematica

CoefficientList[Series[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 21 2015 *)

PARI

x='x + O('x^50); Vec(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, Feb 14 2017

Formula

a(n) = a(n-1) + Sum_{j=0..n-2} A217312(j)*a(n-j).

G.f: 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+3/2) / (32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

Conjecture: (-n+3)*a(n) +3*(2*n-7)*a(n-1) +(-7*n+24)*a(n-2) +2*(-7*n+36)*a(n-3) +2*(11*n-51)*a(n-4) +3*(3*n-23)*a(n-5) +(-10*n+63)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Sep 24 2016