OFFSET
2,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 2..821
FORMULA
From Paul D. Hanna, Nov 22 2024: (Start)
G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x) = x^2 + x^3 + A(A(x)).
(2) A(x) = F(x^2 + x^3) where F(x) = x + F(F(x)^2 + F(x)^3) is the g.f. of A155134.
(3) A(x) = x - Series_Reversion(F(x)) where F(x) is the g.f. of A155134.
(4) F(x) = x + A(F(x)) where F(x) is the g.f. of A155134.
(End)
EXAMPLE
G.f.: A(x) = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 9*x^7 + 20*x^8 + 47*x^9 + 112*x^10 + 274*x^11 + 678*x^12 + ...
where A(x) = x^2 + x^3 + A(A(x)) and
A(A(x)) = x^4 + 2*x^5 + 4*x^6 + 9*x^7 + 20*x^8 + 47*x^9 + 112*x^10 + ...
From Paul D. Hanna, Nov 22 2024: (Start)
RELATED SERIES.
Given F(x) = x + A(F(x)) is the g.f. of A155134, which begins
F(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 207*x^6 + 979*x^7 + 4796*x^8 + 24124*x^9 + 123862*x^10 + ...
then A(x) = F(x^2 + x^3) and F(x - A(x)) = x.
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.34844670790113290932047330524970048405096003118972...
where 1/4 = t^2 + t^3 + A(1/4).
A(t) = 1/5 at t = 0.33447340960120387143777559515829371527354227052793...
where 1/5 = t^2 + t^3 + A(1/5).
A(t) = 1/6 at t = 0.31779916461703610934795588003720521570155887278845...
where 1/6 = t^2 + t^3 + A(1/6).
A(1/3) = 0.19729194074856394185665355090279425985114528328539...
where A(1/3) = 4/27 + A(A(1/3)).
A(1/4) = 0.08627819699468782785875625375630075586019674782414...
where A(1/4) = 5/64 + A(A(1/4)).
A(1/5) = 0.05070917455434681744998768220800119087585037970036...
where A(1/5) = 6/125 + A(A(1/5)).
A(1/6) = 0.03357381499583393562010903064953014653327742866916...
where A(1/6) = 7/216 + A(A(1/6)).
(End)
PROG
(PARI) {a(n)=local(A=x^2+x^3); for(i=1, n, A = x^2 + x^3 + subst(A, x, A+x*O(x^n))); polcoeff(A, n)}
for(n=2, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 24 2012
STATUS
approved