Find the coefficients of $x^{17}$ and $x^{18}$ after expansion and collecting the terms of $(1+x^5+x^7)^{20}$.
To find the coefficients of \(x^{17}\) and \(x^{18}\) in the expansion of \((1 + x^5 + x^7)^{20}\), we need to consider the multinomial expansion and the possible combinations of the exponents that sum to 17 and 18.

1. **Identify the possible combinations for \(x^{17}\):**
   - We need to find the combinations of \(0, 5,\) and \(7\) that sum to 17.
   - The only combination that works is \(5 + 5 + 7 = 17\).

2. **Calculate the number of ways to choose the terms:**
   - We need to choose two \(x^5\) terms, one \(x^7\) term, and the rest \(1\) terms from the 20 terms.
   - The number of ways to do this is given by the multinomial coefficient:
     \[
     \binom{20}{2, 1, 17} = \frac{20!}{2! \cdot 1! \cdot 17!}
     \]
   - Simplifying this:
     \[
     \binom{20}{2, 1, 17} = \frac{20!}{2! \cdot 1! \cdot 17!} = \frac{20 \times 19 \times 18!}{2 \times 1 \times 18!} = \frac{20 \times 19}{2} = 190
     \]

3. **Identify the possible combinations for \(x^{18}\):**
   - We need to find the combinations of \(0, 5,\) and \(7\) that sum to 18.
   - There is no combination of \(0, 5,\) and \(7\) that sums to 18. Therefore, the coefficient of \(x^{18}\) is 0.

Conclusion:
- The coefficient of \(x^{17}\) is 190.
- The coefficient of \(x^{18}\) is 0.

The final answer is \(\boxed{190}\) for \(x^{17}\) and \(\boxed{0}\) for \(x^{18}\).