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EXAMPLE
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G.f.: A(x) = x + x^2 + 4*x^3 + 40*x^4 + 675*x^5 + 16023*x^6 +...
Coefficients of successive iterations of A(x) are as follows:
[1,1,4,40,675,16023,488264,18150408,798091737,40691618165,...];
[1,2,10,101,1658,38327,1144412,41900873,1821202358,...];
[(1),3,18,189,3063,69204,2024472,72937181,3130859853,...];
[1,(4),28,310,5028,111648,3202916,113479712,4806652072,...];
[1,5,(40),470,7715,169475,4777592,166446886,6951781033,...];
[1,6,54,(675),11310,247443,6875376,235657813,9700212066,...];
[1,7,70,931,(16023),351372,9658544,326079187,13225766705,...];
[1,8,88,1244,22088,(488264),13331864,444121464,17753548560,...];
[1,9,108,1620,29763,666423,(18150408),597989364,23574131721,...];
[1,10,130,2065,39330,895575,24428084,(798091737),31060981566,...];
[1,11,154,2585,51095,1186988,32546888,1057515833,(40691618165),...];
where the diagonal of terms in parenthesis form this sequence.
Some explicit expansions:
A(A(x)) = x + 2*x^2 + 10*x^3 + 101*x^4 + 1658*x^5 + 38327*x^6 +...;
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 189*x^4 + 3063*x^5 + 69204*x^6 +...;
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 310*x^4 + 5028*x^5 + 111648*x^6 +...
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