

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
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OFFSET

0,3


LINKS

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


FORMULA

G.f.: 1/(1xx^2/(1xx^2/(1x+x^2*(1M(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: Dfinite with recurrence (n+2)*a(n) +(7*n17)*a(n1) +2*(7*n+17)*a(n2) +(n+22)*a(n3) +(16*n89)*a(n4) +(4*n+23)*a(n5) +3*(n5)*a(n6)=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/(1xx^2/(1xx^2/(1x+x^2*(1(1x(12*x3*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/(1xx^2/(1xx^2/(1x+x^2*(1(1x(12*x3*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



