%I #40 Mar 22 2023 18:11:46

%S 1,3,10,34,117,405,1407,4899,17083,59629,208284,727900,2544751,

%T 8898873,31125138,108881166,380928795,1332824049,4663705782,

%U 16319702046,57109857519,199859075307,699435489795,2447823832671,8566818534141,29982268505595,104933418068332

%N Binomial transform of A054341 and inverse binomial transform of A049027.

%C First column of the Riordan array ((1-2x)/(1+x+x^2),x/(1+x+x^2))^(-1). [_Paul Barry_, Nov 06 2008]

%C Apparently the Motzkin transform of A125176, supposed A125176 is interpreted with offset 0. [_R. J. Mathar_, Dec 11 2008]

%C 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

%H Vincenzo Librandi, <a href="/A059738/b059738.txt">Table of n, a(n) for n = 0..1000</a>

%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%H Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Shattuck/sh36.html">Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries</a>, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%F a(n) = Sum[k=0..n, 2^(n-k)*A026300(n, k) ], where A026300 is the Motzkin triangle. - _Ralf Stephan_, Jan 25 2005 [Corrected by _Philippe Deléham_, Nov 29 2009]

%F a(n)= A126954(n,0). [_Philippe Deléham_, Nov 24 2009]

%F G.f.: 2/(1-5*x+sqrt(1-2*x-3*x^2)). - _Emeric Deutsch_, May 02 2011

%F Recurrence: 2*(n+1)*a(n) = (11*n+5)*a(n-1) - (8*n+5)*a(n-2) - 21*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 11 2012

%F a(n) ~ 3*7^n/2^(n+2). - _Vaclav Kotesovec_, Oct 11 2012

%F G.f.: 1/(1 - 3*x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Nov 19 2021

%t Table[SeriesCoefficient[2/(1-5*x+Sqrt[1-2*x-3*x^2]),{x,0,n}],{n,0,20}]

%o (PARI) x='x+O('x^66); Vec(2/(1-5*x+sqrt(1-2*x-3*x^2))) \\ _Joerg Arndt_, May 06 2013

%K nonn

%O 0,2

%A _John W. Layman_, Feb 09 2001

%E More terms from _Vincenzo Librandi_, May 06 2013