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

%I A140049 #12 Oct 03 2019 04:21:04
%S A140049 1,1,5,55,1005,26601,941863,42372177,2336926665,153927536545,
%T A140049 11869936146891,1055015092106889,106731589524249517,
%U A140049 12163935655214359329,1548324822731892094191,218516875165035215308801,33979477899236956531288977,5790103152487972170694748097
%N A140049 E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
%H A140049 Alois P. Heinz, <a href="/A140049/b140049.txt">Table of n, a(n) for n = 0..282</a>
%F A140049 a(n) = A140054(n+1)/(n+1).
%F A140049 E.g.f.: A(x) = exp(G(x)) where G(x) = e.g.f. of A140055.
%F A140049 E.g.f. satisfies: A(x) = exp( x*A(x) * A(x*A(x)) ).
%F A140049 From _Paul D. Hanna_, Jul 09 2009: (Start)
%F A140049 E.g.f. satisfies: A(x) = exp(x*A(x)*A(x*A(x))).
%F A140049 ...
%F A140049 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then
%F A140049 a(n,m) = Sum_{k=0..n} C(n,k) * m*(n+m)^(k-1) * a(n-k,k).
%F A140049 ...
%F A140049 Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then
%F A140049 L(n) = Sum_{k=1..n} C(n,k) * n^(k-1) * a(n-k,k).
%F A140049 (End)
%e A140049 A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
%e A140049 Log(A(x)) = G(x) = e.g.f. of A140055:
%e A140049 Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
%p A140049 b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*j
%p A140049       *b(j-1, j)*b(n-j, k)*binomial(n-1, j-1), j=1..n))
%p A140049     end:
%p A140049 a:= n-> b(n, n+1):
%p A140049 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 21 2019
%t A140049 m = 18; A[_] = 0;
%t A140049 Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}];
%t A140049 CoefficientList[A[x], x] * Range[0, m-1]! (* _Jean-François Alcover_, Oct 03 2019 *)
%o A140049 (PARI) {a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n+1)}
%o A140049 (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)}
%o A140049 From _Paul D. Hanna_, Jul 09 2009: (Start)
%o A140049 (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))))}
%o A140049 (PARI) /* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
%o A140049 {L(n)=if(n<1,0,sum(k=1,n,binomial(n,k)*n^(k-1)*a(n-k,k)))} (End)
%Y A140049 Cf. A140054, A140055.
%Y A140049 Cf. A162659. [From _Paul D. Hanna_, Jul 09 2009]
%K A140049 nonn
%O A140049 0,3
%A A140049 _Paul D. Hanna_, May 06 2008