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A192338 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+2.

%I A192338 #10 Nov 22 2012 07:51:46
%S A192338 1,2,6,18,54,166,514,1610,5078,16118,51394,164474,527798,1697254,
%T A192338 5466498,17627370,56892246,183742358
%N A192338 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+2.
%C A192338 For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%F A192338 Conjecture: a(n) = 4*a(n-1)+a(n-2)-10*a(n-3)-4*a(n-4). G.f.: x*(2*x^3-3*x^2-2*x+1) / ((x^2+2*x-1)*(4*x^2+2*x-1)). [_Colin Barker_, Nov 22 2012]
%e A192338 The first five polynomials at A157751 and their reductions are as follows:
%e A192338 p0(x)=1 -> 1
%e A192338 p1(x)=2+x -> 2+x
%e A192338 p2(x)=4+2x+x^2 -> 6+3x
%e A192338 p3(x)=8+4x+4x^2+x^3 -> 18+11x
%e A192338 p4(x)=16+8x+12x^2+4x^3+x^4 -> 54+37x.
%e A192338 From these, we read
%e A192338 A192338=(1,2,6,18,54,...) and A192339=(0,1,3,11,37,...)
%t A192338 q[x_] := x + 2;
%t A192338 p[0, x_] := 1;
%t A192338 p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /;
%t A192338   n > 0  (* polynomials defined at A157751 *)
%t A192338 Table[Simplify[p[n, x]], {n, 0, 5}]
%t A192338 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192338    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192338 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,16}]
%t A192338 Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]
%t A192338 (* A192338 *)
%t A192338 Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]
%t A192338 (* A192339 *)
%Y A192338 Cf. A192232, A192339.
%K A192338 nonn
%O A192338 1,2
%A A192338 _Clark Kimberling_, Jun 28 2011