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

1, 2, 6, 18, 54, 166, 514, 1610, 5078, 16118, 51394, 164474, 527798, 1697254, 5466498, 17627370, 56892246, 183742358

Offset: 1

Author

Clark Kimberling, Jun 28 2011

Keywords

• nonn

Comments

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

Examples

			The first five polynomials at A157751 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=2+x -> 2+x
p2(x)=4+2x+x^2 -> 6+3x
p3(x)=8+4x+4x^2+x^3 -> 18+11x
p4(x)=16+8x+12x^2+4x^3+x^4 -> 54+37x.
From these, we read
A192338=(1,2,6,18,54,...) and A192339=(0,1,3,11,37,...)
		

Crossrefs

Cf. A192232, A192339.

Programs

Mathematica

q[x_] := x + 2;
p[0, x_] := 1;
p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /;
  n > 0  (* polynomials defined at A157751 *)
Table[Simplify[p[n, x]], {n, 0, 5}]
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, 0,16}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]
(* A192338 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]
(* A192339 *)

Formula

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]