A100099 An inverse Chebyshev transform of x/(1-2x).

0, 1, 2, 7, 16, 46, 110, 295, 720, 1870, 4612, 11782, 29224, 73984, 184102, 463687, 1156000, 2902870, 7245020, 18161170, 45356736, 113576596, 283765132, 710118262, 1774619616, 4439253196, 11095532840, 27749232700, 69363052600

Offset: 0

Author

Paul Barry, Nov 04 2004

Keywords

• easy
• nonn

Comments

Image of x/(1-2*x) under the transform g(x)->(1/sqrt(1-4*x^2))*g(x*c(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

Programs

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 *)

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