OFFSET
0,2
COMMENTS
Hankel transform is (-3)^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)/(1-4x) 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
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
FORMULA
G.f.: (1/sqrt(1-4*x^2))*(1+x*c(x^2))/(1-4*x*c(x^2)) with c(x) = (1-sqrt(1-4*x))/(2*x).
a(n) = Sum_{k=0..n} A061554(n,k)*4^k. - Philippe Deléham, Dec 04 2009
Recurrence: 4*n*a(n) = (17*n + 8)*a(n-1) + 2*(8*n - 33)*a(n-2) - 68*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 5*17^n/4^(n+1). - Vaclav Kotesovec, Oct 19 2012
MATHEMATICA
CoefficientList[Series[(1/Sqrt[1-4x^2])*(1+x*(1-Sqrt[1-4*x^2])/(2*x^2))/(1-4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 11 2007
STATUS
approved