OFFSET
1,3
COMMENTS
It appears that a(A219608(n)) is odd for n >= 1, and that the only other odd term is a(2) = 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( A(x) - 2*A(x)^2 * A'(x) ).
(2) A(A(x)) = x + 2*A(A(x))^2 * A'(A(x)).
(3) R(R(x)) = x - 2*x^2 * A'(R(x)), where A(R(x)) = x.
a(n) ~ c * d^n * n! * n^alpha, where d = 1.3534821142256737694364485294..., alpha = 2.7625501039589..., c = 0.0101323266748276... - Vaclav Kotesovec, Sep 07 2024
EXAMPLE
G.f.: A(x) = x + x^2 + 5*x^3 + 40*x^4 + 414*x^5 + 5096*x^6 + 71465*x^7 + 1113432*x^8 + 18964415*x^9 + 349252420*x^10 + ...
where x = A( A(x) - 2*A(x)^2 * A'(x) ).
RELATED SERIES.
Let R(x) be the series reversion of A(x) so that R(A(x)) = x, then
R(x) = x - x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2240*x^6 - 30759*x^7 - 475116*x^8 - 8081145*x^9 - 149243380*x^10 + ...
where R(x) = A(x) - 2*A(x)^2 * A'(x).
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 90*x^5 + 933*x^6 + 11420*x^7 + 158862*x^8 + 2453874*x^9 + ...
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = -polcoeff( subst(Ax, x, Ax - 2*Ax^2*Ax')/2, #A-1); ); H=Ax; A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 06 2024
STATUS
approved