0,3

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)) ).

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

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! +...

(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)}

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

Cf. A140054, A140055.

Cf. A162659. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]

Sequence in context: A135861 A141361 A006150 this_sequence A130031 A119399 A158690

Adjacent sequences: A140046 A140047 A140048 this_sequence A140050 A140051 A140052

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), May 06 2008