A140054 E.g.f. A(x) satisfies: A( x*exp(-A(x)) ) = x.

1, 2, 15, 220, 5025, 159606, 6593041, 338977416, 21032339985, 1539275365450, 130569297615801, 12660181105282668, 1387510663815243721, 170295099173001030606, 23224872340978381412865, 3496270002640563444940816, 577651124287028261031912609, 104221856744783499072505465746

Offset

1,2

Comments

Unsigned version of A087962.

Not the same as A178533.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..282 (first 75 terms from Paul D. Hanna)

Formula

E.g.f. A(x) satisfies:

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

(2) A(x) = x*exp( A(x)*exp( A(A(x))*exp( A(A(A(x)))*exp( ...)))) (infinite exponential tower).

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

then

(3.a) A_{n+1}(x) = A( A_{n}(x) ) = A_{n}(x) * exp( A_{n+2}(x) );

(3.b) A_{n}(x) = x*exp( Sum_{k=2..n+1} A_{k}(x) ).

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

a(n)=n!*T(n,1), T(n,m)=m/n*sum(T(n-m,k)*n^k/k!,k,1,n-m), n>m, T(n.n)=1. [Vladimir Kruchinin, May 06 2012]

Example

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! +...
Iterations 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)))) ).
...

Maple

b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
      b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n-1, n)*n:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*b[j - 1, j]*j*b[n - j, k]* Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n - 1, n]*n;
a /@ Range[1, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

Prog

(PARI) {a(n)=local(A=x); for(i=0, n, A=serreverse(x*exp(-A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 20, print1(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)}
for(n=1, 20, print1(a(n), ", "))
(Maxima) A(n, m):=if n=m then 1 else m/n*sum(A(n-m, k)*n^k/k!, k, 1, n-m);
makelist(n!*A(n, 1), n, 1, 10); [Vladimir Kruchinin, May 06 2012]

Crossrefs

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

Cf. A178533.

Sequence in context: A135860 A178533 A087962 * A099085 A078365 A207037

Adjacent sequences: A140051 A140052 A140053 * A140055 A140056 A140057

Keyword

nonn

Author

Paul D. Hanna, May 03 2008

Status

approved