A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629, 208284, 727900, 2544751, 8898873, 31125138, 108881166, 380928795, 1332824049, 4663705782, 16319702046, 57109857519, 199859075307, 699435489795, 2447823832671, 8566818534141, 29982268505595, 104933418068332

Offset: 0

Author

John W. Layman, Feb 09 2001

Keywords

• nonn

Comments

First column of the Riordan array ((1-2x)/(1+x+x^2),x/(1+x+x^2))^(-1). [Paul Barry, Nov 06 2008]

Apparently the Motzkin transform of A125176, supposed A125176 is interpreted with offset 0. [R. J. Mathar, 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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
• Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
• 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.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Programs

Mathematica

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

PARI

x='x+O('x^66); Vec(2/(1-5*x+sqrt(1-2*x-3*x^2))) \\ Joerg Arndt, May 06 2013

Formula

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]

a(n)= A126954(n,0). [Philippe Deléham, Nov 24 2009]

G.f.: 2/(1-5*x+sqrt(1-2*x-3*x^2)). - Emeric Deutsch, May 02 2011

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

a(n) ~ 3*7^n/2^(n+2). - Vaclav Kotesovec, Oct 11 2012

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

Extensions

More terms from Vincenzo Librandi, May 06 2013