%I #14 Nov 22 2024 06:25:24

%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

%F D-finite with recurrence n*a(n) +4*(-3*n+2)*a(n-1) +(45*n-58)*a(n-2) +2*(-27*n+46)*a(n-3) +20*(n-2)*a(n-4)=0. - _R. J. Mathar_, Nov 22 2024

%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