Questions: The generating function of the sequence (an) is (f(x)=fracx^2(1+2 x)^2(1-x) .) Use it to find a formula for (an). Hint: (left(frac11+2 xright)^prime=frac-2(1+2 x)^2)

$The generating function of the sequence (an) is (f(x)=fracx^2(1+2 x)^2(1-x) .) Use it to find a formula for (an). Hint: (left(frac11+2 xright)^prime=frac-2(1+2 x)^2)$

Transcript text: 1. (10 points) The generating function of the sequence $\left(a_{n}\right)_{n}$ is \[ f(x)=\frac{x^{2}}{(1+2 x)^{2}(1-x)} . \] Use it to find a formula for $a_{n}$. Hint: \[ \left(\frac{1}{1+2 x}\right)^{\prime}=\frac{-2}{(1+2 x)^{2}} \]

Solution

Solution Steps

To find a formula for $a_n$ using the generating function $f(x) = \frac{x^2}{(1+2x)^2(1-x)}$, we can follow these steps:

Decompose the generating function: Break down $f(x)$ into simpler parts that we can handle separately.
Use known series expansions: Utilize known series expansions for $\frac{1}{1-x}$ and $\frac{1}{(1+2x)^2}$.
Combine the series: Multiply the series expansions together to get the coefficient of $x^n$, which will give us $a_n$.

Step 1: Generating Function

The generating function given is

\[ f(x) = \frac{x^2}{(1 + 2x)^2(1 - x)}. \]

Step 2: Series Expansion

We expand $f(x)$ into a power series around $x = 0$:

\[ f(x) = x^2 - 3x^3 + 9x^4 - 23x^5 + 57x^6 - 135x^7 + 313x^8 - 711x^9 + O(x^{10}). \]

Step 3: Coefficients Extraction

From the series expansion, we can extract the coefficients corresponding to $x^n$: