%I #28 Sep 08 2022 08:45:29
%S 1,-1,4,-7,22,-46,130,-295,790,-1870,4864,-11782,30148,-73984,187534,
%T -463687,1168870,-2902870,7293640,-18161170,45541492,-113576596,
%U 284470564,-710118262,1777323772,-4439253196,11105933440,-27749232700,69403169200
%N a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*(-2)^(n-k).
%C Hankel transform is 3^n. In general, for r >= 0, the sequence given by Sum_{k=0..n} binomial(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+2*x) under the Chebyshev mapping 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.
%C Second binomial transform is A026641. - _Philippe Deléham_, Mar 14 2007
%C Signed version of A100098. - _Philippe Deléham_, Nov 25 2007
%H Vincenzo Librandi, <a href="/A127361/b127361.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1/sqrt(1-4*x^2))(1+x*c(x^2))/(1+2*x*c(x^2)), with c(x) = (1 - sqrt(1-4*x))/(2*x).
%F a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - _Philippe Deléham_, Nov 25 2007
%F a(n) = Sum_{k=0..n} A061554(n,k)*(-2)^k. - _Philippe Deléham_, Dec 04 2009
%F 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 30 2012
%F a(n) ~ (-1)^n * 5^n / 2^(n+1). - _Vaclav Kotesovec_, Feb 13 2014
%p a:=n->add(binomial(n,floor(k/2))*(-2)^(n-k),k=0..n): seq(a(n),n=0..30); # _Muniru A Asiru_, Feb 18 2019
%t CoefficientList[Series[(1/Sqrt[1-4*x^2])*(1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) /(1+2*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (PARI) my(x='x+O('x^30)); Vec( (1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ) \\ _G. C. Greubel_, Feb 17 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1+2*x-Sqrt(1-4*x^2))/(2*Sqrt(1-4*x^2)*(1+x-Sqrt(1-4*x^2))) )); // _G. C. Greubel_, Feb 17 2019
%o (Sage) ((1+2*x-sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*(1+x-sqrt(1-4*x^2))) ).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 17 2019
%Y Cf. A000108, A026641, A061554.
%K easy,sign
%O 0,3
%A _Paul Barry_, Jan 11 2007
%E More terms from _Vincenzo Librandi_, Feb 15 2014