The next term in the sequence is:

>! **22**

Because:

>! These are the **[chapter numbers](https://harrypotter.fandom.com/wiki/List_of_chapters_in_the_Harry_Potter_novels)** which share their name with the part of the book title that follows '*Harry Potter and...*' (in order of the books' publication):  
>!  
>! Book 1 - **none**: *no chapter called 'The Philosopher's Stone'*;  
>! Book 2 - Chapter **16**: 'The Chamber of Secrets';  
>! Book 3 - **none**: *no chapter called 'The Prisoner of Azkaban'*;  
>! Book 4 - Chapter **16**: 'The Goblet of Fire';  
>! Book 5 - Chapter **5**: 'The Order of the Phoenix';  
>! Book 6 - Chapter **9**: 'The Half-Blood Prince';  
>! **Book 7 - Chapter 22: 'The Deathly Hallows'.**

**EDIT:** Since some commenters wish to understand the thought process behind my solving this puzzle, here's roughly how I approached it:

>! 1) There are 7 terms in the sequence, and there are 7 HP books - let's assume that each term in the sequence maps to one of the books...  
>!  
>! 2) Why are the two null entries written as 'None' rather than '0'? This probably means we're not counting something, otherwise the OP would have just used a zero character;  
>!  
>! 3) What other numbers are relevant to books - chapter numbers, to start...  
>!  
>! 4) Look up the chapters of these numbers in the relevant books;  
>!  
>! 5) *GASP AS YOU SPOT THE CONNECTION!!*