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A192340 Constant term of the reduction of n-th polynomial at A158985 by x^2->x+1.

%I A192340 #6 Mar 30 2012 18:57:34
%S A192340 1,3,19,1091,4270307,65975813893475,15748607358316275150858234851,
%T A192340 897339846665475127909937786392825941994036757434025817827,
%U A192340 2913308988276889310145046342161059349226587591969604604068795694857825566722967409631885309325418272374141705507555
%N A192340 Constant term of the reduction of n-th polynomial at A158985 by x^2->x+1.
%C A192340 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%e A192340 The first three polynomials at A158985 and their reductions are as follows:
%e A192340 p0(x)=1+x -> 1+x
%e A192340 p1(x)=2+2x+x^2 -> 3+3x
%e A192340 p2(x)=5+8x+8x^2+4x^3+x^4 -> 19+27x.
%e A192340 From these, we read
%e A192340 A192340=(1,3,19,...) and A192341=(1,3,27,...)
%t A192340 q[x_] := x + 1;
%t A192340 p[0, x_] := x + 1;
%t A192340 p[n_, x_] := 1 + p[n - 1, x]^2 /; n > 0  (* polynomials defined at A158985 *)
%t A192340 Table[Expand[p[n, x]], {n, 0, 4}]
%t A192340 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),  x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192340 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 9}]
%t A192340 Table[Coefficient[Part[t, n], x, 0], {n, 1, 9}]
%t A192340 (* A192340 *)
%t A192340 Table[Coefficient[Part[t, n], x, 1], {n, 1, 9}]
%t A192340 (* A192341 *)
%Y A192340 Cf. A192232, A192341.
%K A192340 nonn
%O A192340 1,2
%A A192340 _Clark Kimberling_, Jun 28 2011