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%I #20 Dec 15 2019 18:23:04
%S 1,5,22,95,406,1730,7360,31295,133030,565430,2403172,10213670,
%T 43408444,184486580,784069252,3332296895,14162266630,60189642830,
%U 255806000260,1087175537570,4620496103956,19637108580380,83457711731152,354695275386470,1507454921406556
%N a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*4^(n-k).
%C Hankel transform is (-3)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-4x) 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="/A127360/b127360.txt">Table of n, a(n) for n = 0..300</a>
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F G.f.: (1/sqrt(1-4*x^2))*(1+x*c(x^2))/(1-4*x*c(x^2)) with c(x) = (1-sqrt(1-4*x))/(2*x).
%F a(n) = Sum_{k=0..n} A061554(n,k)*4^k. - _Philippe Deléham_, Dec 04 2009
%F Recurrence: 4*n*a(n) = (17*n + 8)*a(n-1) + 2*(8*n - 33)*a(n-2) - 68*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 5*17^n/4^(n+1). - _Vaclav Kotesovec_, Oct 19 2012
%t CoefficientList[Series[(1/Sqrt[1-4x^2])*(1+x*(1-Sqrt[1-4*x^2])/(2*x^2))/(1-4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%Y Cf. A107430. - _Philippe Deléham_, Sep 16 2009
%Y Cf. A061554.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 11 2007
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