%I #28 Sep 22 2025 16:00:59

%S 1,5,20,81,332,1372,5702,23793,99576,417664,1754866,7383204,31096466,

%T 131084954,552969854,2334012425,9856336324,41639407776,175971686398,

%U 743888534968,3145439344550,13302946909338,56272308538682

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

%C Hankel transform is A165204.

%H Iain Fox, <a href="/A165203/b165203.txt">Table of n, a(n) for n = 0..1595</a> (first 201 terms from Vincenzo Librandi)

%F G.f. (for offset 1): (1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2)).

%F a(n) = (A165201(n) - 0^n) + A165201(n+1).

%F Conjecture: (n+1)*(5*n-31)*a(n) +(5*n^2+74*n+62)*a(n-1) +(-285*n^2+ 1072*n-757)*a(n-2) +(695*n^2-3674*n+4206)*a(n-3) +2*(45*n-74)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Dec 11 2011

%F a(n) ~ (18/sqrt(5)-8) * (2+sqrt(5))^(n+2). - _Vaclav Kotesovec_, Feb 01 2014

%t Rest[CoefficientList[Series[(1+x)*((1-x)*Sqrt[1-4*x]+5*x-1)/(2*(1-4*x-x^2)), {x, 0, 30}], x]] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%o (PARI) first(n) = x='x+O('x^(n+1)); Vec((1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2))) \\ _Iain Fox_, Feb 27 2018

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))^3/(x*(8*x^2 - (1-Sqrt(1-4*x))^3)) )); // _G. C. Greubel_, Jul 18 2019

%o (SageMath) ((1+x)*(1-sqrt(1-4*x))^3/(x*(8*x^2 - (1-sqrt(1-4*x))^3)) ).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 18 2019

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 07 2009