A202020 Number of 4-colored Motzkin paths of length n with no peaks at level 1.

1, 4, 16, 68, 305, 1428, 6914, 34368, 174438, 900392, 4712034, 24944268, 133335497, 718664500, 3901458106, 21313500576, 117081025390, 646328535800, 3583680016616, 19949056745160, 111447034042634

Offset: 0

Author

José Luis Ramírez Ramírez, Dec 08 2011

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A135334.

Programs

Mathematica

CoefficientList[Series[(2*x^2-4*x+1-Sqrt[12*x^2-8*x+1])/(2*x^4-8*x^3+4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

PARI

z='z+O('z^50); Vec((2*z^2-4*z+1-sqrt(12*z^2-8*z+1))/(2*z^4-8*z^3+ 4*z^2)) \\ G. C. Greubel, Mar 29 2017

Formula

G.f.: (2*z^2-4*z+1 - sqrt(12*z^2-8*z+1))/(2*z^4-8*z^3+4 z^2).

Conjecture: 2(n+2)*a(n) -4*(5n+4)*a(n-1) +3*(19n-2)*a(n-2) +4*(11-14n)*a(n-4) +12*(n-1)*a(n-4)=0. - R. J. Mathar, Dec 18 2011

a(n) ~ 18*6^(n+3/2)/(49*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012