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A140049
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E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
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1
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1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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)) ).
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)
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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! +...
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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)}
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)
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CROSSREFS
| Cf. A140054, A140055.
Cf. A162659. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]
Sequence in context: A141361 A203013 A006150 * A130031 A119399 A177557
Adjacent sequences: A140046 A140047 A140048 * A140050 A140051 A140052
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), May 06 2008
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