OFFSET
0,3
COMMENTS
Image of x/(1-2x-x^2) 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)).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: x*sqrt(1-4*x^2)*(sqrt(1-4*x^2)+2*x)/((1-4*x^2)*(1-8*x^2)).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*A000129(n-2k)}.
Conjecture: (-n+2)*a(n) +(-n+3)*a(n-1) +4*(3*n-7)*a(n-2) +4*(3*n-10)*a(n-3) +32*(-n+3)*a(n-4) +32*(-n+4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012
Recurrence: (n-2)*a(n) = 4*(3*n-7)*a(n-2) - 32*(n-3)*a(n-4). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 2^((3*n-3)/2). - Vaclav Kotesovec, Feb 12 2014
a(2*n) = 8^n/(2*sqrt(2)) - 2^n * (2*n-1)!! * hypergeom([1, n+1/2], [n+1], 1/2)/(4*n!), a(2*n+1) = 8^n. - Vladimir Reshetnikov, Oct 13 2016
MATHEMATICA
CoefficientList[Series[x*Sqrt[1-4*x^2]*(Sqrt[1-4*x^2]+2*x)/((1-4*x^2)*(1-8*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Nov 03 2004
STATUS
approved