A257290 Number of 3-Motzkin paths of length n with no level steps at even level.

1, 0, 1, 3, 11, 39, 140, 504, 1823, 6621, 24144, 88380, 324699, 1197045, 4427565, 16427385, 61129025, 228103185, 853399640, 3200710680, 12032399045, 45332769075, 171148151095, 647412581643, 2453529142471, 9314461044639, 35419207688050, 134894888442714, 514506926871927

Offset

0,4

Links

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

Formula

a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.

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

a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

Example

For n=3 we have 3 paths: UH1D, UH2D, UH3D.

Mathematica

CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

Prog

(PARI) x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017

Crossrefs

Cf. A090345, A025266.

Sequence in context: A289834 A007482 A134760 * A371758 A281482 A132889

Adjacent sequences: A257287 A257288 A257289 * A257291 A257292 A257293

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 20 2015

Status

approved