A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

1, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368

Offset: 1

Author

Clark Kimberling, Jun 29 2011

Keywords

• nonn

Comments

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).
		

Crossrefs

Cf. A192232, A192380, A192381.

Programs

Mathematica

q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 2014