Hailey, The verb "forget" takes the infinitive as object when you are looking to the future from a point in the past. The infinitive is "to shut". I forgot to shut the window.... He should have shut the window. The window is open. "Forget" takes the "ing" or gerund form of the verb as object when you are looking back at the past. In this case you used the past gerund "having shut". I forgot having shut the window.... means: I forgot that I had shut the window. The window is closed.

"I forgot to shut the window" means that you left the window open as you went away. forgot + infinitive with to- You may say "I don´t remember having shut the window" if you are not sure whether you shut it or not But you can´t say "I forgot having shut the window" because you can´t use the past simple tense + present perfect continuos tense-