OFFSET
0,3
COMMENTS
Image of (1-x)/(1-2x) under the transform g(x)->(1/sqrt(1-4xx^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).
Transform of the Jacobsthal numbers A001045(n+1) under the Riordan array (c(x^2),xc(x^2)). Hankel transform is 3^n. - Paul Barry, Oct 01 2007
Unsigned version of A127361. - Philippe Deléham, 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)-6x+3)/(2*(2-5x)*(1-4x^2));
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*(2^(n-2k) + 0^(n-2k))/2.
From Paul Barry, Oct 01 2007: (Start)
G.f.: (1+2x+3*sqrt(1-4x^2))/(4-2x-20x^2);
a(n) = Sum_{k=0..floor((n+1)/2)} (C(n,k) - C(n,k-1))*A001045(n-2k+1). (End)
Conjecture: 2*n*a(n) + (-5*n+4)*a(n-1) + 2*(-4*n+3)*a(n-2) + 20*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 22 2012
a(n) ~ 5^n / 2^(n+1). - Vaclav Kotesovec, Feb 08 2014
MATHEMATICA
CoefficientList[Series[Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]-6*x+3)/(2*(2-5*x)*(1-4*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 04 2004
STATUS
approved