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

1,3

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.

Links

Table of n, a(n) for n=1..26.

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

Example

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,...).

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 *)

Crossrefs

Cf. A192232, A192380, A192381.

Sequence in context: A256401 A109292 A275571 * A117474 A294665 A323139

Adjacent sequences: A192376 A192377 A192378 * A192380 A192381 A192382

Keyword

nonn

Author

Clark Kimberling, Jun 29 2011

Status

approved