%I #9 Oct 14 2019 06:55:13

%S 1,4,42,764,20400,731862,33397168,1867950856,124680486816,

%T 9733666171850,874978919826264,89437471672859532,10289414670501314608,

%U 1320997962702267801070,187894667581541881127640,29426125555003596239544848,5046809953516305090792395328

%N E.g.f.: A(x) = G(G(x)) where G(x) = x*exp(A(x)) such that G( x*exp(-G(x)) ) = x and G(x) is the e.g.f. of A140054.

%H Alois P. Heinz, <a href="/A140055/b140055.txt">Table of n, a(n) for n = 1..282</a>

%e E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...

%e x*exp(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...

%e where G(x) = x*exp(A(x)) satisfies G(G(x)) = A(x).

%p b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*

%p b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Aug 21 2019

%t m = 20; G[_] = 0;

%t Do[G[x_] = x Exp[G[G[x]]] + O[x]^(m+1) // Normal, {m}];

%t CoefficientList[G[G[x]]/x + O[x]^m, x]*Range[m]! (* _Jean-François Alcover_, Oct 14 2019 *)

%o (PARI) {a(n)=local(A=x);for(i=0,n,A=x*exp(subst(A,x,A+x*O(x^n))));n!*polcoeff(subst(A,x,A),n)}

%Y Cf. A140054 (x*exp(A(x))).

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 03 2008