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
Keywords
Links
- Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Programs
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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
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Mathematica
CoefficientList[y/.AsymptoticSolve[y-1-x(1-y+y^2)^3/y==0,y->1,{x,0,24}][[1]],x] -
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
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