A347953 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.

1, 1, 2, 8, 35, 171, 882, 4744, 26286, 149045, 860596, 5042968, 29913676, 179270434, 1083794310, 6601817952, 40479778395, 249646876065, 1547539929810, 9637085582640, 60260786147261, 378212395786511, 2381767469829332, 15045137488662048, 95304451461770250

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Formula

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

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

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

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

Maple

cx := (1-sqrt(1-4*x))/2/x ;
tx := 2/sqrt(3*x)*sin( 1/3*arcsin(sqrt(27*x/4))) ;
gf := 1/subs(x=-x*tx^3, cx) ;
taylor(%, x=0, 40) ;
gfun[seriestolist](%) ; # R. J. Mathar, Jul 20 2023

Mathematica

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

Prog

(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

Crossrefs

Cf. A000108, A001764, A166135.

Sequence in context: A011791 A259757 A371826 * A030847 A030929 A030881

Adjacent sequences: A347950 A347951 A347952 * A347954 A347955 A347956

Keyword

nonn

Author

Alexander Burstein, Nov 02 2021

Status

approved