A140049 E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).

1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801, 33979477899236956531288977, 5790103152487972170694748097

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..282

Formula

a(n) = A140054(n+1)/(n+1).

E.g.f.: A(x) = exp(G(x)) where G(x) = e.g.f. of A140055.

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

From Paul D. Hanna, Jul 09 2009: (Start)

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

...

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} C(n,k) * m*(n+m)^(k-1) * a(n-k,k).

...

Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then

L(n) = Sum_{k=1..n} C(n,k) * n^(k-1) * a(n-k,k).

(End)

Example

A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
Log(A(x)) = G(x) = e.g.f. of A140055:
Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...

Maple

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

Mathematica

m = 18; A[_] = 0;
Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Oct 03 2019 *)

Prog

(PARI) {a(n)=local(A=x); for(i=0, n, A=serreverse(x*exp(-A+x*O(x^n)))); n!*polcoeff(A, n+1)}
(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+1)}
From Paul D. Hanna, Jul 09 2009: (Start)
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, binomial(n, k)*m*(n+m)^(k-1)*a(n-k, k))))}
(PARI) /* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
{L(n)=if(n<1, 0, sum(k=1, n, binomial(n, k)*n^(k-1)*a(n-k, k)))} (End)

Crossrefs

Cf. A140054, A140055.

Cf. A162659. [From Paul D. Hanna, Jul 09 2009]

Sequence in context: A371316 A006150 A373500 * A300589 A130031 A336289

Adjacent sequences: A140046 A140047 A140048 * A140050 A140051 A140052

Keyword

nonn

Author

Paul D. Hanna, May 06 2008

Status

approved