A054341 Row sums of triangle A054336 (central binomial convolutions).

1, 2, 5, 12, 30, 74, 185, 460, 1150, 2868, 7170, 17904, 44760, 111834, 279585, 698748, 1746870, 4366460, 10916150, 27287944, 68219860, 170541252, 426353130, 1065853432, 2664633580, 6661479944, 16653699860, 41633878200, 104084695500, 260210401530, 650526003825

Offset: 0

Author

Wolfdieter Lang, Mar 13 2000

Keywords

• easy
• nonn

Comments

a(n) = # Dyck (n+1)-paths all of whose components are symmetric. A strict Dyck path is one with exactly one return to ground level (necessarily at the end). Every nonempty Dyck path is expressible uniquely as a concatenation of one or more strict Dyck paths, called its components. - David Callan, Mar 02 2005

a(n) = # 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height. Example: a(2)=5 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BB, BR, RB, RR, and UD. - Emeric Deutsch, Jun 07 2011

Inverse Chebyshev transform of the second kind applied to 2^n. This maps g(x) -> c(x^2)g(xc(x^2)). - Paul Barry, Sep 14 2005

Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007

Inverse binomial transform of A059738. - Philippe Deléham, Nov 24 2009

Examples

			a(4) = 30, the upper left term of M^4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2513 (first 201 terms from Vincenzo Librandi)
• Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Crossrefs

Cf. A000108, A054336.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, 1,
      b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Jan 23 2024

Mathematica

CoefficientList[Series[2/(1-4*x+Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

a(n) = Sum_{m=0..n} A054336(n, m).

G.f.: 1/(1-2*x-x^2*c(x^2)), where c(x) = g.f. for Catalan numbers A000108.

From_Paul Barry_, Sep 14 2005: (Start)

G.f.: c(x^2)/(1-2*x*c(x^2));

a(n) = Sum_{k=0..n} binomial(n,(n-k)/2)*(1 + (-1)^(n+k))*2^k*(k+1)/(n+k+2). (End)

G.f.: 2/(1-4*x+sqrt(1-4*x^2)). - Ira M. Gessel, Oct 27 2013

a(n) = A127358(n+1) - 2*A127358(n). - Philippe Deléham, Mar 02 2007

a(n) = A126075(n,0). - Philippe Deléham, Nov 24 2009

a(n) = Sum_{k=0..n} A053121(n,k)*2^k. - Philippe Deléham, Nov 28 2009

From Gary W. Adamson, Sep 07 2011: (Start)

a(n) is the upper left term of M^n, M is an infinite square production matrix as follows:

2, 1, 0, 0, 0, ...

1, 0, 2, 0, 0, ...

0, 1, 0, 1, 0, ...

0, 0, 1, 0, 1, ...

0, 0, 0, 1, 0, ...

... (End)

Conjecture: 2*(n+1)*a(n) +5*(-n-1)*a(n-1) +8*(-n+2)*a(n-2) +20*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012

a(n) ~ 3 * 5^n / 2^(n+2). - Vaclav Kotesovec, Feb 13 2014