A127362 - OEIS

%I #17 Nov 22 2024 08:04:33

%S 1,-2,8,-24,84,-272,920,-3040,10180,-33840,112968,-376224,1254696,

%T -4181088,13939248,-46459584,154873860,-516229040,1720795880,

%U -5735921440,19119861304,-63732624672,212442552528,-708140901184,2360471473384,-7868234639072

%N a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-3)^(n-k).

%C Hankel transform is 4^n. In general, for r>=0, the sequence given by sum{k=0..n, C(n,floor(k/2))*(-r)^(n-k)} has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+3x) 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.

%H Vincenzo Librandi, <a href="/A127362/b127362.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1+3*x*c(x^2)).

%F D-finite with recurrence 3*n*a(n) +2*(5*n-3)*a(n-1) +4*(-3*n+1)*a(n-2) +40*(-n+2)*a(n-3)=0. - _R. J. Mathar_, Nov 15 2012

%F a(n) ~ (-1)^n * 2^(n+1) * 5^n / 3^(n+1). - _Vaclav Kotesovec_, Feb 08 2014

%F G.f.: 1/(-1+2*x+2*sqrt(1-4*x^2)). - _Vaclav Kotesovec_, Feb 08 2014

%t CoefficientList[Series[1/(-1+2*x+2*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%K easy,sign

%O 0,2

%A _Paul Barry_, Jan 11 2007