If you just want the next line after a pattern, this sed command will work
sed -n -e '/pattern/{n;p;}'
-n supresses output (quiet mode);
-e denotes a sed command (not required in this case);
/pattern/ is a regex search for lines containing the literal combination of the characters pattern (Use /^pattern$/ for line consisting of only of “pattern”;
n replaces the pattern space with the next line;
p prints;
For example:
seq 10 | sed -n -e '/5/{n;p;}'
Note that the above command will print a single line after every line containing pattern. If you just want the first one use sed -n -e '/pattern/{n;p;q;}'. This is also more efficient as the whole file is not read.
This strictly sed command will print all lines after your pattern.
sed -n '/pattern/,${/pattern/!p;}
Formatted as a sed script this would be:
/pattern/,${
    /pattern/!p
}
Here’s a short example:
seq 10 | sed -n '/5/,${/5/!p;}'
/pattern/,$ will select all the lines from pattern to the end of the file.
{} groups the next set of commands (c-like block command)
/pattern/!p; prints lines that doesn’t match pattern. Note that the ; is required in early versions, and some non-GNU, of sed. This turns the instruction into a exclusive range - sed ranges are normally inclusive for both start and end of the range.
To exclude the end of range you could do something like this:
sed -n '/pattern/,/endpattern/{/pattern/!{/endpattern/d;p;}}
/pattern/,/endpattern/{
    /pattern/!{
        /endpattern/d
        p
    }
}
/endpattern/d is deleted from the “pattern space” and the script restarts from the top, skipping the p command for that line.
Another pithy example:
seq 10 | sed -n '/5/,/8/{/5/!{/8/d;p}}'
If you have GNU sed you can add the debug switch:
seq 5 | sed -n --debug '/2/,/4/{/2/!{/4/d;p}}'
Output:
SED PROGRAM:
  /2/,/4/ {
    /2/! {
      /4/ d
      p
    }
  }
INPUT:   'STDIN' line 1
PATTERN: 1
COMMAND: /2/,/4/ {
COMMAND: }
END-OF-CYCLE:
INPUT:   'STDIN' line 2
PATTERN: 2
COMMAND: /2/,/4/ {
COMMAND:   /2/! {
COMMAND:   }
COMMAND: }
END-OF-CYCLE:
INPUT:   'STDIN' line 3
PATTERN: 3
COMMAND: /2/,/4/ {
COMMAND:   /2/! {
COMMAND:     /4/ d
COMMAND:     p
3
COMMAND:   }
COMMAND: }
END-OF-CYCLE:
INPUT:   'STDIN' line 4
PATTERN: 4
COMMAND: /2/,/4/ {
COMMAND:   /2/! {
COMMAND:     /4/ d
END-OF-CYCLE:
INPUT:   'STDIN' line 5
PATTERN: 5
COMMAND:     /2/,/4/ {
COMMAND:     }
END-OF-CYCLE: