A192338 - OEIS

%I #10 Nov 22 2012 07:51:46

%S 1,2,6,18,54,166,514,1610,5078,16118,51394,164474,527798,1697254,

%T 5466498,17627370,56892246,183742358

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

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

%F 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 The first five polynomials at A157751 and their reductions are as follows:

%e p0(x)=1 -> 1

%e p1(x)=2+x -> 2+x

%e p2(x)=4+2x+x^2 -> 6+3x

%e p3(x)=8+4x+4x^2+x^3 -> 18+11x

%e p4(x)=16+8x+12x^2+4x^3+x^4 -> 54+37x.

%e From these, we read

%e A192338=(1,2,6,18,54,...) and A192339=(0,1,3,11,37,...)

%t q[x_] := x + 2;

%t p[0, x_] := 1;

%t p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /;

%t   n > 0  (* polynomials defined at A157751 *)

%t Table[Simplify[p[n, x]], {n, 0, 5}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t    x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,16}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]

%t (* A192338 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]

%t (* A192339 *)

%Y Cf. A192232, A192339.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jun 28 2011