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%I #12 Jan 29 2017 02:19:29
%S 0,1,4,15,59,243,1034,4501,19920,89281,404184,1844789,8477571,
%T 39183625,182010366,849115811,3976405347,18684473203,88060677880,
%U 416162484693,1971567963673,9361218368921,44539107835094,212308063827055,1013779444844754,4848597239921803
%N Composition of the binomial transform of Fibonacci numbers and the Catalan transform of Fibonacci numbers.
%H G. C. Greubel, <a href="/A219312/b219312.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="http://repository.wit.ie/201/1/CatalanTrans.pdf">A Catalan transform and related transformations on integer sequences</a>, pp. 20-22.
%F G.f.: (sqrt(5*x-1) - sqrt(x-1))/(2*((x-1)*sqrt(5*x-1) - x*sqrt(x-1))).
%F a(n) ~ 5^(n+5/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 19 2013
%t CoefficientList[Series[(Sqrt[5*x-1] - Sqrt[x-1])/(2*((x-1)*Sqrt[5*x-1] - x*Sqrt[x-1])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 19 2013 *)
%o (PARI) Vec((sqrt(5*x-1) - sqrt(x-1))/(2*((x-1)*sqrt(5*x-1) - x*sqrt(x-1))) + O(x^25)) \\ _G. C. Greubel_, Jan 28 2017
%Y Cf. A000045.
%K easy,nonn
%O 0,3
%A _Arkadiusz Wesolowski_, Nov 17 2012
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