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A165201 Expansion of 1/(1-x*c(x)^3), c(x) the g.f. of A000108.

%I A165201 #20 Sep 08 2022 08:45:47
%S A165201 1,1,4,16,65,267,1105,4597,19196,80380,337284,1417582,5965622,
%T A165201 25130844,105954110,447015744,1886996681,7969339643,33670068133,
%U A165201 142301618265,601586916703,2543852427847,10759094481491,45513214057191
%N A165201 Expansion of 1/(1-x*c(x)^3), c(x) the g.f. of A000108.
%C A165201 Hankel transform is A165202. Essentially the same as A026674.
%H A165201 Vincenzo Librandi, <a href="/A165201/b165201.txt">Table of n, a(n) for n = 0..300</a>
%F A165201 G.f.: (1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2)).
%F A165201 a(n) = (1/2)*Sum_{k=0..n} C(2k,k)*F(3(n-k)+1)/(1-2k) + (1/2)*(F(3n-2) + 2*0^n).
%F A165201 Conjecture: n*(n-3)*a(n) +2*(-4*n^2+15*n-10)*a(n-1) +(15*n^2-69*n+80)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) =0. - _R. J. Mathar_, Nov 15 2011
%F A165201 a(n) ~ 1/10*(3*sqrt(5)-5)*(sqrt(5)+2)^n. - _Vaclav Kotesovec_, Oct 20 2012
%t A165201 CoefficientList[Series[(1-3*x-2*x^2+(1-x)*Sqrt[1-4*x])/(2*(1-4*x-x^2)), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A165201 (PARI) my(x='x+O('x^30)); Vec((1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))) \\ _G. C. Greubel_, Jul 18 2019
%o A165201 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-3*x-2*x^2 + (1-x)*Sqrt(1-4*x))/(2*(1-4*x-x^2)) )); // _G. C. Greubel_, Jul 18 2019
%o A165201 (Sage) ((1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 18 2019
%o A165201 (GAP) List([0..30], n-> (1/2)*(2*0^n + Fibonacci(3*n-2) + Sum([0..n], j-> Binomial(2*j, j)*Fibonacci(3*(n-j)+1)/(1-2*j) ))); # _G. C. Greubel_, Jul 18 2019
%Y A165201 Cf. A000108, A026674, A165202.
%K A165201 easy,nonn
%O A165201 0,3
%A A165201 _Paul Barry_, Sep 07 2009