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EXAMPLE
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Coefficients in the initial iterations of F(x) = x*(1+x)^2 begin:
[(1),0,0,0,0,0,0,0,0,0,...];
[1,(2),1,0,0,0,0,0,0,0,...];
[1,4,(10),18,23,22,15,6,1,0,...];
[1,6,27,(102),333,960,2472,5748,12150,23388,...];
[1,8,52,300,(1578),7692,35094,150978,615939,2393628,...];
[1,10,85,660,4790,(32920),215988,1360638,8265613,48585702,...];
[1,12,126,1230,11385,101010,(864813),7178700,57976074,456783888,...];
[1,14,175,2058,23163,251832,2660028,(27405798),276215313,2729492562,...];
[1,16,232,3192,42308,544600,6842220,84191772,(1017153322),12088865688,...];
[1,18,297,4680,71388,1061712,15463512,221228244,3115739358,(43271534196),...];
where the terms enclosed in parenthesis form this sequence.
Some explicit expansions:
F(x) = x + (2)x^2 + x^3 ;
F(F(x)) = x + 4x^2 + (10)x^3 + 18x^4 + 23x^5 + 22x^6 + 15x^7 +...;
F(F(F(x))) = x + 6x^2 + 27x^3 + (102)x^4 + 333x^5 + 960x^6 +...;
F(F(F(F(x)))) = x + 8x^2 + 52x^3 + 300x^4 + (1578)x^5 + 7692x^6 +...;
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