OFFSET
0,2
COMMENTS
Hankel transform is (-1)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k)} has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-2x) under the Chebyshev mapping g(x) -> (1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
FORMULA
G.f.: (1/sqrt(1 - 4x^2))(1 + x*c(x^2))/(1 - 2*x*c(x^2)).
a(n) = Sum_{k=0..n} A061554(n,k)*2^k. - Philippe Deléham, Dec 04 2009
From Gary W. Adamson, Sep 07 2011: (Start)
a(n) is the sum of top row terms of M^n, M is an infinite square production matrix as follows:
2, 1, 0, 0, 0, ...
1, 0, 1, 0, 0, ...
0, 1, 0, 1, 0, ...
0, 0, 1, 0, 1, ...
0, 0, 0, 1, 0, ...
... (End)
Conjecture: 2*n*a(n) + (-5*n-4)*a(n-1) + 2*(-4*n+13)*a(n-2) + 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 30 2012
a(n) ~ 3 * 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 13 2014
EXAMPLE
a(3) = 21 = (12 + 6 + 2 + 1), where the top row of M^3 = (12, 6, 2, 1).
MATHEMATICA
Table[Sum[Binomial[n, Floor[k/2]]2^(n-k), {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Jun 03 2012 *)
CoefficientList[Series[(1 + 2*x - Sqrt[1 - 4*x^2])/(2*Sqrt[1 - 4*x^2]*(x - 1 + Sqrt[1 - 4*x^2])), {x, 0, 50}], x] (* G. C. Greubel, May 22 2017 *)
PROG
(PARI) x='x+O('x^50); Vec((1 + 2*x - sqrt(1 - 4*x^2))/(2*sqrt(1 - 4*x^2)*(x - 1 + sqrt(1 - 4*x^2)))) \\ G. C. Greubel, May 22 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 11 2007
STATUS
approved