OFFSET
0,3
COMMENTS
Image of x/(1-2x) under the transform g(x)->(1/sqrt(1-4x^2)g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))A(x/(1+x^2).
Hankel transform is A125905(n-1), the alternating sign version of A001353. - Paul Barry, Nov 25 2007
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: sqrt(1-4x^2)(sqrt(1-4x^2)+4x-1)/(2(5x-2)(4x^2-1)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*(2^(n-2*k)-0^(n-2*k))/2.
a(n) = Sum_{k=0..n} C(n,floor(k/2))*A001045(n-k). - Paul Barry, Nov 25 2007
Conjecture: 2n*a(n) +(-13n+16)*a(n-1) +4(3n-8)*a(n-2) +4(13n-29)*a(n-3) +80(3-n)*a(n-4)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 01 2014
MATHEMATICA
CoefficientList[Series[Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]+4*x-1)/(2*(5*x-2)*(4*x^2-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 04 2004
STATUS
approved