[seqfan] A278332 Needs Extending

[Paul Hanna]

SeqFans,
      Would someone like to extend sequence   https://oeis.org/A278332 ?
The terms begin: 1, 2, 9, 88, 905, 12666, 220297, ...
Below I give the description for your convenience.
Thanks,
     Paul

The e.g.f. of A278332 equals the limit of the average of all permutations
of the compositions of the functions x*exp(x^k), for k=1..n, as n increases.

Generating method.
Define F(n,x) as the average of the sum over all n! permutations of the
compositions of x*exp(x^k) for k=1..n, then the e.g.f. of this sequence is
the limit of the functions F(n,x) as n grows.

Examples of some initial functions F(n,x) are as follows.
The symbol  o  is to denote composition.

At n=1, F(1,x) = x*exp(x) ;
F(1,x) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! + 6*x^6/6! +...

At n=2,
F(2,x) = (1/2!)*([x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x)]) ;

F(2,x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 425*x^5/5! + 3486*x^6/6! +...

At n=3,
F(3,x) = (1/3!)*([x*exp(x) o x*exp(x^2) o x*exp(x^3)] + [x*exp(x^2) o
x*exp(x) o x*exp(x^3)] + [x*exp(x) o x*exp(x^3) o x*exp(x^2)] + [x*exp(x^3)
o x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x^3) o x*exp(x)] +
[x*exp(x^3) o x*exp(x^2) o x*exp(x)]) ;

F(3,x) = x + 2*x^2/2! + 9*x^3/3! + 88*x^4/4! + 785*x^5/5! + 9426*x^6/6! +...

etc.