A257300 Number of Motzkin paths of length n with no peaks at level 2.

1, 1, 2, 4, 8, 17, 38, 88, 210, 514, 1285, 3270, 8447, 22100, 58455, 156077, 420153, 1139155, 3108095, 8527675, 23514124, 65127571, 181111940, 505487115, 1415502195, 3975790024, 11197966459, 31619946886, 89496047586, 253858251337, 721531869889, 2054639741185

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-M(x)))), where M(x) is the g.f. of Motzkin numbers A001006.

a(n) ~ 3^(n+7/2) / (50*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

Conjecture: n*a(n) +(-5*n+3)*a(n-1) +6*(n)*a(n-2) +2*(n-9)*a(n-3) +6*(-n+4)*a(n-4) +(n-6)*a(n-5) +3*(-n+3)*a(n-6)=0. - R. J. Mathar, Sep 24 2016

Example

For n=4 we have 8 paths: HHHH, UDUD, UHDH, HUHD, UHHD, UDHH, HUDH and HHUD.

Mathematica

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

Prog

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

Crossrefs

Cf. A089372.

Sequence in context: A119685 A025276 A006461 * A229202 A003007 A086615

Adjacent sequences: A257297 A257298 A257299 * A257301 A257302 A257303

Keyword

nonn

Author

José Luis Ramírez Ramírez, Apr 20 2015

Status

approved