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A192386 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 3
1, 0, 8, 8, 96, 224, 1408, 4608, 22784, 86016, 386048, 1548288, 6676480, 27467776, 116490240, 484409344, 2040135680, 8521777152, 35786063872, 149761818624, 628140015616, 2630784909312, 11028578435072, 46205266558976, 193656954814464 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+5).  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..25.

FORMULA

Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (16*x^4+8*x^3-12*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)=3+x+3x^2 -> 8+4x

p(3,x)=12x+4x^2+4x^3 -> 8+32x

p(4,x)=9+6x+31x^2+10x^3+5x^4 -> 96+96x.

From these, read A192386=(1,0,8,8,96,...) and A192387=(0,2,4,32,96,...)

MATHEMATICA

q[x_] := x + 1; d = Sqrt[x + 5];

p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* suggested by 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[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];

Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192386 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192387 *)

Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192388 *)

CROSSREFS

Cf. A192232, A083087.

Sequence in context: A082798 A286068 A228071 * A119932 A270118 A270150

Adjacent sequences:  A192383 A192384 A192385 * A192387 A192388 A192389

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified April 26 09:52 EDT 2019. Contains 322472 sequences. (Running on oeis4.)