G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 267*x^7 + 1041*x^8 + 4168*x^9 + 17047*x^10 + 70902*x^11 + 298967*x^12 + 1275141*x^13 + 5491504*x^14 + 23846271*x^15 + ...
RELATED SERIES.
We can illustrate the formulas with the following related expansions.
(1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 + ...
which equals A(x^3) - x^2 + x^4/A(x^3), as can be seen from
x^4/A(x^3) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 + ...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 435*x^8 + 1715*x^9 + ...
x^4/A(x)^3 = x - 3*x^2 - 4*x^4 - 9*x^5 - 30*x^6 - 115*x^7 - 435*x^8 - 1713*x^9 + ...
(2) Let F(x) be the g.f. of A370440, which begins
F(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + ...
where F(x)^3 = F( x^3 + 3*x^2*F(x)^2 ),
then the series reversion of F(x) begins
A(x^3) - x^2 + x^4/A(x^3) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 + ...
(3) Let G(x) be the g.f. of A264228, which begins
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 +...
where G(x)^3 = G( x^3/(1 - 3*x) ),
then the series reversion of G(x) begins
-x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) = x^2/(x + x^2 + x^3 + x^4 - x^6 - x^7 + 2*x^9 + 3*x^10 - 6*x^12 - 9*x^13 + 20*x^15 + 30*x^16 - 71*x^18 - 110*x^19 + 267*x^21 + 419*x^22 - 1041*x^24 +...).
SPECIFIC VALUES.
A(1/4.834464) = 0.349644497578571280258023712232522068793519739...
A(1/5) = 0.29940801195429552263938628184744484915469836164855...
A(1/6) = 0.21539123666426270273178791857213676628593723946879...
A(1/7) = 0.17414937372444126736977770687571455113383911571251...
A(1/8) = 0.14713126344900776621336355426627444003268957268553...
A(1/5^3) = 0.00806504925055020701973761348380106375185943151538...
A(1/6^3) = 0.00465126435780731657600811126033650347236250831668...
A(1/7^3) = 0.00292400175440295890949208907819991271975334925594...
which may be used to verify that the formula
A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2
holds for these specific values.
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