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EXAMPLE
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The first three polynomials at A158983 and their reductions are as follows:
p0(x)=2+x -> 2+x
p1(x)=5+4x+x^2 -> 7+5x
p2(x)=26+40x+26x^2+8x^3+x^4 -> 100+95x.
From these, we read
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MATHEMATICA
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q[x_] := x + 2;
p[0, x_] := x + 2;
p[n_, x_] := 1 + p[n - 1, x]^2 /; n > 0 (* polynomials defined at A158983 *)
Table[Expand[p[n, x]], {n, 0, 4}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 9}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 9}]
Table[Coefficient[Part[t, n], x, 1], {n, 1, 9}]
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