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 can break down the problem into manageable steps.

### Step 1: Rewrite the Expression
We start by rewriting the expression:
\[ (1 + x^2 + x^4) \left(1 - \frac{1}{x^2}\right)^{18} \]

### Step 2: Expand the Binomial
Using the Binomial Theorem, we can expand \( \left(1 - \frac{1}{x^2}\right)^{18} \):
\[ \left(1 - \frac{1}{x^2}\right)^{18} = \sum_{r=0}^{18} \binom{18}{r} (-1)^r \left(\frac{1}{x^2}\right)^r = \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{-2r} \]

### Step 3: Combine with \( (1 + x^2 + x^4) \)
Now we multiply this expansion by \( (1 + x^2 + x^4) \):
\[ (1 + x^2 + x^4) \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{-2r} \]
Distributing \( (1 + x^2 + x^4) \) gives:
\[ \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{-2r} + \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{2 - 2r} + \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{4 - 2r} \]

### Step 4: Identify Terms Contributing to \( x^{-2} \)
We need to find the terms in the above sums that contribute to \( x^{-2} \).

1. From the first sum \( \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{-2r} \):
- We need \( -2r = -2 \) which gives \( r = 1 \).
- Contribution: \( \binom{18}{1} (-1)^1 = -18 \).

2. From the second sum \( \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{2 - 2r} \):
- We need \( 2 - 2r = -2 \) which gives \( 2r = 4 \) or \( r = 2 \).
- Contribution: \( \binom{18}{2} (-1)^2 = \binom{18}{2} = \frac{18 \times 17}{2} = 153 \).

3. From the third sum \( \sum_{r=0}^{18} \binom{18}{r} (-1)^r x^{4 - 2r} \):
- We need \( 4 - 2r = -2 \) which gives \( 2r = 6 \) or \( r = 3 \).
- Contribution: \( \binom{18}{3} (-1)^3 = -\binom{18}{3} = -\frac{18 \times 17 \times 16}{3 \times 2 \times 1} = -816 \).

### Step 5: Combine Contributions
Now we combine all contributions:
\[ \text{Total Coefficient} = -18 + 153 - 816 = -681 \]

### Final Answer
Thus, the coefficient of \( x^{-2} \) in the given expression is:
\[ \boxed{-681} \]