A257104 Number of Motzkin paths of length n with no peaks at level 3.

1, 1, 2, 4, 9, 21, 50, 121, 297, 738, 1854, 4704, 12044, 31097, 80919, 212098, 559718, 1486480, 3971285, 10668975, 28812589, 78192989, 213179869, 583703909, 1604685870, 4428216295, 12263271557, 34074271966, 94972933448, 265486492798, 744177020705, 2091359021671, 5891579293777, 16634993650629, 47069839690554

Offset

0,3

Links

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

Formula

G.f.: 1/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-M(x))))), where M(x) is the g.f. of Motzkin numbers A001006.

a(n) ~ 3^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 27 2015

Conjecture: D-finite with recurrence (-n+2)*a(n) +(7*n-17)*a(n-1) +2*(-7*n+17)*a(n-2) +(n+22)*a(n-3) +(16*n-89)*a(n-4) +(-4*n+23)*a(n-5) +3*(n-5)*a(n-6)=0. - R. J. Mathar, Sep 24 2016

Example

For n=4 we have 9 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH, HHUD and UUDD

Mathematica

CoefficientList[Series[1/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 27 2015 *)

Prog

(PARI) x='x+O('x^50); Vec(1/(1-x-x^2/(1-x-x^2/(1-x+x^2*(1-(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)))))) \\ G. C. Greubel, Apr 08 2017

Crossrefs

Cf. A089372, A257300.

Sequence in context: A091964 A092423 A238438 * A318008 A199410 A091600

Adjacent sequences: A257101 A257102 A257103 * A257105 A257106 A257107

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 27 2015

Status

approved