Formula for A120014(n) -- Simplified

[Paul D. Hanna]

Max, and Seqfans, 
    Using the lovely formula for the n-th term of the m-th power of
Catalan function: 
[x^n] Catalan(x)^m = m*C(2*n+m-1,n)/(n+m), 
I have found the simplest form yet for A120014: 
 
a(n) = n^(n-1) - Sum_{k=1..n-1}
n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!/n! 
 
Thus, the e.g.f. of A120014 does involve the LambertW function as
suspected 
(I am still working on obtaining that e.g.f.).  
 
And it is possible that the above formula can be simplified still. 
Any insights are welcome! 
 
If anyone wishes to know of my derivation for the formula, 
I will supply upon request. 
    Paul 
------------------------------------------------ 
A120014 
Coefficients of x^n in the n-th self-composition of the g.f. of A120009, 
so that: 
a(n) = [x^n] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1. 
 
1, 2, 9, 60, 530, 5892, 79681, 1276760, 23729310, 502780580, 11974950746,

 
COMMENT 
a(n) is divisible by n for n>=1; a(n)/n = A120016(n). 
 
FORMULA 
a(n) = [x^n] x*((1-n+n^2) - n^2*(n+1)*x - n*(1-(n+2)*x)*C(x)
)/(1-n+n^2*x)^2, 
where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108). 
 
EXAMPLE  
Successive self-compositions of F(x), the g.f. of A120009, begin: 
F(x) = (1)x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +... 
F(F(x)) = x + (2)x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +... 
F(F(F(x))) = x + 3x^2 + (9)x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +... 
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (60)x^4 + 192x^5 + 360x^6 +... 
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + (530)x^5 +1955x^6 +... 
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + (5892)x^6 +... 
 
(PARI) 
a(n)=n^(n-1)-sum(k=1,n-1,n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!)/n!