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A109190 Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n. 2
1, 0, 2, 2, 8, 16, 46, 114, 310, 822, 2238, 6094, 16764, 46308, 128650, 358862, 1005056, 2824416, 7962122, 22508350, 63792424, 181219680, 515905018, 1471593638, 4205280902, 12037415526, 34510499066, 99083855234, 284870069780 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) with steps u=(1,1), d=(1,-1) and h=(1,0).

Column 0 of A109189.

The substitution x->x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform), yields the g.f. of (-1)^n*A006355(n). - R. J. Mathar, Nov 10 2008

Apparently also the number of grand Motzkin paths of length n that avoid flat steps at level 0. - David Scambler, Jul 04 2013

Motzkin contexts such that along the path from the root to the hole there are only binary nodes. - Pierre Lescanne, Nov 11 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: (sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2).

G.f.: 1/(1-2x^2*M(x)), M(x) the g.f. of the Motzkin numbers A001006. - Paul Barry, Mar 02 2010

Conjecture: n*a(n) +(3-4*n)*a(n-1) +3*(1-n)*a(n-2) +2*(7*n-15)*a(n-3) +12*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 09 2012

a(n) ~ 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014

EXAMPLE

a(3) = 2 because we have uhd and dhu.

MAPLE

g:=(sqrt(1-2*z-3*z^2)-z)/(1-2*z-4*z^2): gser:=series(g, z=0, 33): 1, seq(coeff(gser, z^n), n=1..30);

MATHEMATICA

CoefficientList[Series[(Sqrt[1-2*x-3*x^2]-x)/(1-2*x-4*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 03 2014 *)

PROG

(PARI) x='x+O('x^55); Vec((sqrt(1-2*x-3*x^2)-x)/(1-2*x-4*x^2)) \\ Altug Alkan, Nov 11 2015

CROSSREFS

Cf. A001006, A006355, A109189.

Sequence in context: A228797 A052970 A220589 * A016120 A188115 A085542

Adjacent sequences: A109187 A109188 A109189 * A109191 A109192 A109193

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 21 2005

STATUS

approved

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Last modified February 5 12:39 EST 2023. Contains 360084 sequences. (Running on oeis4.)