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A257104 Number of Motzkin paths of length n with no peaks at level 3. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 08:32 EST 2020. Contains 338762 sequences. (Running on oeis4.)