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A059738
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Binomial transform of A054341 and inverse binomial transform of A049027.
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8
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1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629, 208284, 727900, 2544751, 8898873, 31125138, 108881166, 380928795, 1332824049, 4663705782
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| First column of the Riordan array ((1-2x)/(1+x+x^2),x/(1+x+x^2))^(-1). [From Paul Barry (pbarry(AT)wit.ie), Nov 06 2008]
Apparently the Motzkin transform of A125176, supposed A125176 is interpreted with offset 0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 11 2008]
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors. Example: a(3)=34 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 1 path of shape UHD. - Emeric Deutsch, May 02 2011
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REFERENCES
| Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
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LINKS
| J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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FORMULA
| a(n) = Sum[k=0..n, 2^(n-k)*A026300(n, k) ], where A026300 is the Motzkin triangle. - R. Stephan, Jan 25 2005 [Corrected by Philippe DELEHAM, Nov 29 2009]
a(n)= A126954(n,0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 24 2009]
G.f.=2/[1-5z+sqrt(1-2z-3z^2)]. - Emeric Deutsch, May 02 2011
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CROSSREFS
| Sequence in context: A113300 A007052 A048580 * A094832 A071725 A026016
Adjacent sequences: A059735 A059736 A059737 * A059739 A059740 A059741
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KEYWORD
| nonn,changed
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Feb 09 2001
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