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A219314 Composition of the inverse binomial transform of Fibonacci numbers and the Catalan transform of Fibonacci numbers.

%I A219314 #20 Apr 12 2023 11:13:25
%S A219314 0,1,0,3,3,13,26,77,192,529,1412,3873,10603,29315,81318,226763,634627,
%T A219314 1782637,5022840,14193457,40211105,114191159,324981030,926720807,
%U A219314 2647513282,7576475383,21716189676,62336237007,179182653117,515717424109,1486119467026
%N A219314 Composition of the inverse binomial transform of Fibonacci numbers and the Catalan transform of Fibonacci numbers.
%H A219314 Fung Lam, <a href="/A219314/b219314.txt">Table of n, a(n) for n = 0..2000</a>
%H A219314 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.
%H A219314 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%F A219314 G.f.: ((1+2*x)*sqrt(1-2*x-3*x^2) - 1 + x + 2*x^2)/(2*(1+x)*(1-2*x-4*x^2)).
%F A219314 Asymptotics: a(n) ~ 3^(n+2)*5/(8*sqrt(3*Pi*n^3)). - _Fung Lam_, Apr 07 2014
%F A219314 Conjecture: n*a(n) -2*n*a(n-1) +11*(-n+2)*a(n-2) +4*(2*n-5)*a(n-3) +8*(5*n-17)*a(n-4) +24*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jun 14 2016
%F A219314 Conjecture: n*(5*n-7)*a(n) -4*(5*n^2-12*n+6)*a(n-1) -(15*n^2-11*n-30) *a(n-2) +2*(35*n^2-119*n+66)*a(n-3) +12*(n-3)*(5*n2)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%Y A219314 Cf. A000045, A219312.
%K A219314 easy,nonn
%O A219314 0,4
%A A219314 _Arkadiusz Wesolowski_, Nov 17 2012