A140055 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.

1, 4, 42, 764, 20400, 731862, 33397168, 1867950856, 124680486816, 9733666171850, 874978919826264, 89437471672859532, 10289414670501314608, 1320997962702267801070, 187894667581541881127640, 29426125555003596239544848, 5046809953516305090792395328

Offset: 1

Author

Paul D. Hanna, May 03 2008

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
x*exp(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
where G(x) = x*exp(A(x)) satisfies G(G(x)) = A(x).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..282

Crossrefs

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

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
      b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Mathematica

m = 20; G[_] = 0;
Do[G[x_] = x Exp[G[G[x]]] + O[x]^(m+1) // Normal, {m}];
CoefficientList[G[G[x]]/x + O[x]^m, x]*Range[m]! (* Jean-François Alcover, Oct 14 2019 *)

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)}