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MATHEMATICA
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c[n_] := 3 n; (* *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 3; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]
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,
30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192307 *)
Table[Coefficient[Part[t, n]/3, x, 0], {n, 1, 30}] (* A190062 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192308 *)
Table[Coefficient[Part[t, n]/3, x, 1], {n, 1, 30}] (* A122491 *)
(* by Peter Moses, Jun 20 2011 *)
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