A138020 G.f. satisfies A(x) = sqrt( (1 + 2*x*A(x)) / (1 - 2*x*A(x)) ).

1, 2, 6, 24, 110, 544, 2828, 15232, 84246, 475648, 2730068, 15882240, 93438540, 554967040, 3323125528, 20039827456, 121597985254, 741871845376, 4548193111428, 28004975116288, 173113004348580, 1073893324357632, 6683288759506856, 41715337804120064

Offset: 0

Author

Paul D. Hanna, Feb 28 2008

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A078531, A151374.

Programs

Maple

A138020 := proc(n)
    option remember ;
    if n < 5 then
        op(n+1,[1,2,6,24,110]) ;
    else
        4*(-55*n^3 +231*n^2 -263*n +51)*procname(n-2) -16*(n-3)*(n-4)*(5*n-1)*procname(n-4) ;
        -%/n/(n+1)/(5*n-11)
    end if;
end proc:
seq(A138020(n),n=0..30) ; # R. J. Mathar, Sep 27 2024

Mathematica

CoefficientList[y/.AsymptoticSolve[y^2-1-2x(y+y^3) ==0,y->1,{x,0,23}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)

PARI

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

Formula

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

From Alexander Burstein, Nov 26 2021: (Start)

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

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

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

From Seiichi Manyama, Dec 22 2024: (Start)

a(n) = (2^n/(n+1)) * Sum_{k=0..n} binomial(n/2+k-1/2,k) * binomial(n/2+1/2,n-k).

a(n) = 2^n * Sum_{k=0..n} binomial(n,k) * binomial(n/2+k+1/2,n)/(n+2*k+1). (End)