G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 43*x^6 + 144*x^7 +...
where:
A(A(x)) = A(x) + A(x)^3/x + A(x)^6/x^3 + A(x)^10/x^6 + A(x)^15/x^10 +...
...
Let q = A(x)/x, then g.f. A(x) satisfies:
A(A(x)) = A(x)*(1-xq)/(1-xq^2) + A(x)^2*(1-xq)(1-xq^3)/((1-xq^2)(1-xq^4)) + A(x)^3*(1-xq)(1-xq^3)(1-xq^5)/((1-xq^2)(1-xq^4)(1-xq^6)) +...
Explicitly,
A(A(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 324*x^6 + 1380*x^7 + 6137*x^8 + 28348*x^9 + 135549*x^10 + 669406*x^11 + 3408490*x^12 +...
Related expansions are:
A(x)^3/x = x^2 + 3*x^3 + 9*x^4 + 28*x^5+ 90*x^6 + 300*x^7 +...
A(x)^6/x^3 = x^3 + 6*x^4 + 27*x^5 + 110*x^6 + 429*x^7 +...
A(x)^10/x^6 = x^4 + 10*x^5 + 65*x^6 + 350*x^7 + 1700*x^8 +...
A(x)^15/x^10 = x^5 + 15*x^6 + 135*x^7 + 950*x^8 + 5775*x^9 +...
A(x)^21/x^15 = x^6 + 21*x^7 + 252*x^8 + 2275*x^9 + 17199*x^10 +...
...
Let G(x) satisfy A(G(x)) = x, then
A(x) = x + x^3/G(x) + x^6/G(x)^3 + x^10/G(x)^6 + x^15/G(x)^10 +...
where:
G(x) = x - x^2 - x^6 - 4*x^7 - 9*x^8 - 26*x^9 - 129*x^10 - 537*x^11 - 1961*x^12 - 9088*x^13 - 44722*x^14 - 199057*x^15 -...
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