You can use awk:
awk '/pattern/{c=5;next} !(c&&c--)' file
Basically: We are decreasing the integer c on every row of input. We are printing lines when c is 0. *(see below) Note: c will be automatically initialized with 0 by awk upon it's first usage.
When the word pattern is found, we set c to 5 which makes c--<=0 false for 5 lines and makes awk not print those lines.
* We could bascially use c--<=0 to check if c is less or equal than 0. But when there are many(!) lines between the occurrences of the word pattern, c could overflow. To avoid that, oguz ismail suggested to implement the check like this:
!(c&&c--)
This will check if c is trueish (greater zero) and only then decrement c. c will never be less than 0 and therefore not overflow. The inversion of this check !(...) makes awk print the correct lines.
Side-note: Normally you would use the word regexp if you mean a regular expression, not pattern.