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A109190
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Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n. (A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).).
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1
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 10 2008]
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FORMULA
| G.f.=[sqrt(1-2z-3z^2)-z]/(1-2z-4z^2).
G.f.: 1/(1-2x^2*M(x)), M(x) the g.f. of the Motzkin numbers A001006. [From Paul Barry (pbarry(AT)wit.ie), Mar 02 2010]
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EXAMPLE
| a(3)=2 because we have uhd and dhu.
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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);
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CROSSREFS
| Cf. A109189.
Sequence in context: A098273 A192305 A052970 * A016120 A188115 A085542
Adjacent sequences: A109187 A109188 A109189 * A109191 A109192 A109193
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 21 2005
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