grammar - What is the formal expression of 住んでんだ

I came to this colloquial expression 住んでんだ from One Piece.

Answer

It's contracted from 住んでいるのだ:

1.  住んでいるのだ
2.  住んでいるんだ　　(contract の to ん)
3.  住んでるんだ　　　(contract ている to てる)
4. *住んでんんだ　　　(contract る to ん)
5.  住んでんだ　　　　(reduce double ん to single ん)

Step 4 isn't a valid sequence; the *んん is not allowed here, so you're forced to reduce the invalid sequence *んんだ to んだ when you contract る to ん before んだ.

I marked step 4 with a * to show that it's not a valid sequence.