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G.f.: A(x) = 1/C(-x*T(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.
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%I #28 Jul 20 2023 15:15:03

%S 1,1,2,8,35,171,882,4744,26286,149045,860596,5042968,29913676,

%T 179270434,1083794310,6601817952,40479778395,249646876065,

%U 1547539929810,9637085582640,60260786147261,378212395786511,2381767469829332,15045137488662048,95304451461770250

%N G.f.: A(x) = 1/C(-x*T(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 and T(x) = 1 + x*T(x)^3 is the g.f. of A001764.

%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.

%F G.f.: A(-x*A(x)^3) = 1/A(x).

%F G.f.: The series reversion of x*A(x)^3 is x*A(-x)^3.

%F G.f.: A(x) satisfies A(x) = 1 + x*(1 - A(x) + A(x)^2)^3/A(x).

%F D-finite with recurrence +4*n*(4*n-1)*(4*n+1)*a(n) +6*(-342*n^3+1233*n^2-1453*n+542)*a(n-1) +243*(n-2)*(33*n^2-123*n+112)*a(n-2) +2187*(n-3)*(3*n-4)*(3*n-8)*a(n-3)=0. - _R. J. Mathar_, Jul 20 2023

%p cx := (1-sqrt(1-4*x))/2/x ;

%p tx := 2/sqrt(3*x)*sin( 1/3*arcsin(sqrt(27*x/4))) ;

%p gf := 1/subs(x=-x*tx^3,cx) ;

%p taylor(%,x=0,40) ;

%p gfun[seriestolist](%) ; # _R. J. Mathar_, Jul 20 2023

%t CoefficientList[y/.AsymptoticSolve[y-1-x(1-y+y^2)^3/y==0,y->1,{x,0,24}][[1]],x]

%o (PARI) seq(n) = {Vec(1/subst((1 - sqrt(1 - 4*x + O(x^2*x^n))) / (2*x), x, -serreverse(x / (1+x)^3 + O(x*x^n))))} \\ _Andrew Howroyd_, Nov 22 2021

%Y Cf. A000108, A001764, A166135.

%K nonn

%O 0,3

%A _Alexander Burstein_, Nov 02 2021