A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.

1, 0, 5, 1, 42, 43, 429, 820, 4861, 12597, 58598, 177859, 732825, 2417416, 9358677, 32256553, 120902914, 426440955, 1571649221, 5610955132, 20497829133, 73645557469, 267803779710, 965384509651, 3502058316337, 12646311635088, 45818284122149

Offset: 1

Author

Clark Kimberling, Jun 29 2011

Keywords

• nonn

Comments

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

Examples

			(See A192352 for a related example.)
		

Crossrefs

Cf. A192232, A192354, A192352.

Programs

Mathematica

q[x_] := x + 1; d = 2;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
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, 30}]
Table[Coefficient[Part[t,n],x,0], {n,1,30}](* A192353 *)
Table[Coefficient[Part[t,n],x,1], {n,1,30}]  (* A192354 *)

Formula

Empirical G.f.: x*(x^3-4*x^2-2*x+1)/((x^2+3*x+1)*(5*x^2-5*x+1)). [Colin Barker, Sep 11 2012]