Find the coefficient of `x^(-2)` in `(1+x^(2)+x^(4)) (1-1/(x^(2)))^(18)`

Answer

To find the coefficient of \( x^{-2} \) in the expression \( (1 + x^2 + x^4) \left(1 - \frac{1}{x^2}\right)^{18} \), we will follow these steps:

### Step 1: Expand \( \left(1 - \frac{1}{x^2}\right)^{18} \)

Using the binomial theorem, we can expand \( \left(1 - \frac{1}{x^2}\right)^{18} \):
\[ \left(1 - \frac{1}{x^2}\right)^{18} = \sum_{k=0}^{18} \binom{18}{k} (-1)^k \left(\frac{1}{x^2}\right)^k = \sum_{k=0}^{18} \binom{18}{k} (-1)^k x^{-2k} \]

### Step 2: Combine with \( (1 + x^2 + x^4) \)

Now we need to multiply this expansion by \( (1 + x^2 + x^4) \):
\[ (1 + x^2 + x^4) \sum_{k=0}^{18} \binom{18}{k} (-1)^k x^{-2k} \]

### Step 3: Identify terms that contribute to \( x^{-2} \)

We need to find the terms from the product that will give us \( x^{-2} \). This can happen in the following ways:
1. From \( 1 \) in \( (1 + x^2 + x^4) \) and the term \( x^{-2} \) from \( \left(1 - \frac{1}{x^2}\right)^{18} \).
2. From \( x^2 \) in \( (1 + x^2 + x^4) \) and the term \( x^{-4} \) from \( \left(1 - \frac{1}{x^2}\right)^{18} \).
3. From \( x^4 \) in \( (1 + x^2 + x^4) \) and the term \( x^{-6} \) from \( \left(1 - \frac{1}{x^2}\right)^{18} \).

### Step 4: Calculate the coefficients

1. **From \( 1 \) and \( x^{-2} \)**:
- The coefficient of \( x^{-2} \) is \( \binom{18}{1} (-1)^1 = -18 \).

2. **From \( x^2 \) and \( x^{-4} \)**:
- The coefficient of \( x^{-4} \) is \( \binom{18}{2} (-1)^2 = \binom{18}{2} = \frac{18 \times 17}{2} = 153 \).

3. **From \( x^4 \) and \( x^{-6} \)**:
- The coefficient of \( x^{-6} \) is \( \binom{18}{3} (-1)^3 = -\binom{18}{3} = -\frac{18 \times 17 \times 16}{3 \times 2 \times 1} = -816 \).

### Step 5: Combine the contributions

Now we combine the contributions:
\[ \text{Total coefficient of } x^{-2} = -18 + 153 - 816 = -681 \]

### Final Answer
The coefficient of \( x^{-2} \) in the expansion is \( -681 \).

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