A138020 - OEIS

%I #12 Sep 27 2024 06:22:55

%S 1,2,6,24,110,544,2828,15232,84246,475648,2730068,15882240,93438540,

%T 554967040,3323125528,20039827456,121597985254,741871845376,

%U 4548193111428,28004975116288,173113004348580,1073893324357632

%N G.f. satisfies A(x) = sqrt( (1 + 2x*A(x)) / (1 - 2x*A(x)) ).

%H Vaclav Kotesovec, <a href="/A138020/b138020.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 2^(n - 1/2) * phi^((5*n + 3)/2)  / (sqrt(Pi) * 5^(1/4) * n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Oct 04 2020

%F From _Alexander Burstein_, Nov 26 2021: (Start)

%F G.f.: A(x) = 1 + 2*x*A(x)*(1 + A(x)^2)/(1 + A(x)).

%F G.f.: A(-x*A(x)^2) = 1/A(x). (End)

%F D-finite with recurrence +n*(n+1)*(5*n-11) *a(n) +4*(-55*n^3 +231*n^2 -263*n +51)*a(n-2) -16*(n-3)*(n-4)*(5*n-1)*a(n-4)=0. - _R. J. Mathar_, Mar 25 2024

%p A138020 := proc(n)

%p     option remember ;

%p     if n < 5 then

%p         op(n+1,[1,2,6,24,110]) ;

%p     else

%p         4*(-55*n^3 +231*n^2 -263*n +51)*procname(n-2) -16*(n-3)*(n-4)*(5*n-1)*procname(n-4) ;

%p         -%/n/(n+1)/(5*n-11)

%p     end if;

%p end proc:

%p seq(A138020(n),n=0..30) ; # _R. J. Mathar_, Sep 27 2024

%t CoefficientList[y/.AsymptoticSolve[y^2-1-2x(y+y^3) ==0,y->1,{x,0,21}][[1]],x] - _Alexander Burstein_, Nov 26 2021

%o (PARI) a(n)=polcoeff((1/x)*serreverse(x*sqrt((1-2*x)/(1+2*x+x^2*O(x^n)))),n)

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 28 2008