Formula for A112319? - Diagonals in Self-Compositions of (x+x^2)

[Paul D. Hanna]

Seqfans, 
     Would someone like to try to find a formula for A112319:  
"Coefficients of x^n in the (n-1)-th self-composition of (x + x^2) for
n>=1."
1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264, 
 
The related triangles A122888 and A122890 seem to indicate that 
A112319 may have a formula involving the Catalan numbers: 
http://www.research.att.com/~njas/sequences/A122890
 
Below I copy A112319 and triangle A122888 for your convenience. 
Other diagonals in self-compositions of (x+x^2) are A112317, A112320.
 
Be it a recurrence, series reversion, etc., any formula for these
diagonals 
could lead to other discoveries involving self-compositions of functions.

     Paul 
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A122888
Triangle, read by rows, where row n lists the coefficients of x^k, 
k=1..2^n, in the n-th self-composition of (x + x^2) for n>=0. 

EXAMPLE  
Triangle begins: 
1; 
1, 1; 
1, 2, 2, 1; 
1, 3, 6, 9, 10, 8, 4, 1; 
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1; 

Multiplying the g.f. of column k by (1-x)^k, k>=1, with leading zeros, 
yields the g.f. of row k in the triangle A122890: 
1; 
0, 1; 
0, 0, 2; 
0, 0, 1, 5; 
0, 0, 0, 10, 14; 
0, 0, 0, 8, 70, 42; 
0, 0, 0, 4, 160, 424, 132; 
0, 0, 0, 1, 250, 1978, 2382, 429; 
0, 0, 0, 0, 302, 6276, 19508, 12804, 1430; ... 

in which the main diagonal is the Catalan numbers 
and the row sums form the factorials. 

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A112319  
Coefficients of x^n in the (n-1)-th self-composition of (x + x^2) for
n>=1. 

1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, 1059312264, 
27715541568, 800423573676, 25289923553700, 867723362137464, 
32128443862364255, 1276818947065793736, 54208515369076658640, 
2448636361058495090816
 
FORMULA  
a(n) = [x^n] F_{n-1}(x) where F_n(x) = F_{n-1}(x+x^2) 
with F_1(x) = x+x^2 and F_0(x)=x for n>=1. 
 
EXAMPLE  
Initial terms in self-compositions of (x+x^2) are: 
F(x) = x + (1)*x^2 
F(F(x)) = x + 2*x^2 + (2)*x^3 + x^4 
F(F(F(x))) = x + 3*x^2 + 6*x^3+ (9)*x^4 +... 
F(F(F(F(x)))) = x + 4*x^2 + 12*x^3 + 30*x^4 + (64)*x^5 +... 
F(F(F(F(F(x))))) = x + 5*x^2 + 20*x^3 + 70*x^4 + 220*x^5 + (630)*x^6 +...
 
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