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Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
2

%I #7 Jan 17 2013 09:05:16

%S 1,0,4,4,36,88,432,1408,5776,20736,80320,297792,1132096,4242304,

%T 16028928,60276736,227287296,855703552,3224482816,12144337920,

%U 45752574976,172339107840,649223532544,2445572276224,9212566081536,34703459811328

%N Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%C To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

%F Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]

%e The first four polynomials p(n,x) and their reductions are as follows:

%e p(0,x)=1 -> 1

%e p(1,x)=x -> x

%e p(2,x)=3+x+x^2 -> 4+2x

%e p(3,x)=9x+3x^2+x^3 -> 4+14x.

%e From these, we read

%e A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)

%t q[x_] := x + 1; d = Sqrt[x + 3];

%t p[n_, x_] := ((x + d)^n + (x - d)^n )/

%t 2 (* similar to polynomials defined at A161516 *)

%t Table[Expand[p[n, x]], {n, 0, 4}]

%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, 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]

%t (* A192348 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

%t (* A192349 *)

%Y Cf. A192232, A192349.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 28 2011