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)}
(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))))} \\ Paul D. Hanna, Jul 09 2009
(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)))} \\ Paul D. Hanna, Jul 09 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 06 2008
STATUS
approved