Calculating Coefficients Using Geometric Series

[bakerconspira]

Homework Statement

Calculate $$[x^n] (1-2x+x^2)^{(-k)}$$

Homework Equations

Just the geometric series.

The Attempt at a Solution

This is what I got so far $$[x^n] (1-2x+x^2)^{(-k)} = [x^n]((x-1)^2)^{(-k)} = [x^n] \frac{1}{(x-1)^{(2k)}}$$
Basically, how do I put this into a sum form? or can I just multiply like this $$\frac{1}{(x-1)^{(2k)}} * \frac{1-x}{1-x} \Rightarrow a_n = \frac{(1-x)}{(x-1)^{(2k)}}$$?

 

[HallsofIvy]

I have no clue what you are asking! "Calculate ...". What do you mean by "calculate" an expression? You say "just the geometric series". What about a geometric series? Are you asked to write that expression in terms of a geometric series?

 

[Gib Z]

Differentiate term by term 2k times:

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$$

NB: Halls, [x^n]f(x) is sometimes used to denote the coefficient of x^n in f(x) .