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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
OFFSET
0,3
LINKS
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
Sequence in context: A091964 A092423 A238438 * A318008 A199410 A091600
KEYWORD
nonn
AUTHOR
STATUS
approved