OFFSET
0,2
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
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.
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) = 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
EXAMPLE
a(4) = 30, the upper left term of M^4.
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Mar 13 2000
STATUS
approved