1,2

Unsigned version of A087962.

E.g.f. satisfies: A(x) = x*exp( A(A(x)) ).

E.g.f. satisfies: A(x) = x*exp( A(x)*exp( A(A(x))*exp( A(A(A(x)))*exp( ...)))) (infinite exponential tower).

E.g.f. satisfies: exp(-A(x)) = G(x) where G(x*G(x)) = exp(-x) and G(-x) = e.g.f. of A087961.

E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...

Related expansions are:

exp(-A(x)) = 1 - x - x^2/2! - 10*x^3/3! - 159*x^4/4! - 3816*x^5/5! -...

A(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...

A(A(A(x))) = x + 6*x^2/2! + 81*x^3/3! + 1776*x^4/4! + 55125*x^5/5! +...

A(A(A(A(x)))) = x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 121080*x^5/5! +...

Self-compositions of A(x) obey the relation illustrated by:

A(x) = x*exp( A(A(x)) );

A(A(x)) = x*exp( A(A(x)) + A(A(A(x))) );

A(A(A(x))) = x*exp( A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) ).

Let A_{n}(x) denote n-th self-composition of e.g.f. A(x) with A_0(x)=x,

then A_{n+1}(x) = A( A_{n}(x) ) = A_{n}(x) * exp( A_{n+2}(x) ).

(PARI) {a(n)=local(A=x); for(i=0, n, A=serreverse(x*exp(-A+x*O(x^n)))); n!*polcoeff(A, n)}

(PARI) {a(n)=local(A=x); for(i=0, n, A=x*exp(subst(A, x, A+x*O(x^n)))); n!*polcoeff(A, n)}

Cf. A087962 (A(-x)), A087961 (exp(-A(-x))), A140055 (A(A(x))).

Adjacent sequences: A140051 A140052 A140053 this_sequence A140055 A140056 A140057

Sequence in context: A132493 A135860 A087962 this_sequence A099085 A078365 A090301

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), May 03 2008