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

0,3

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A005043, A217312.

Sequence in context: A206119 A085748 A317097 * A052806 A329672 A218552

Adjacent sequences: A252351 A252352 A252353 * A252355 A252356 A252357

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 20 2015

Status

approved