The German word for “invitation” is “Einladung” (f.), “invited” is “eingeladen”, “to invite” is “einladen” and “inviting” is “einladend”:

– “Hast du die Einladung zu seiner Geburtstagsfeier bekommen?” – “Did you receive the invitation to his birthday party?”
– “Nein, ich bin leider nicht eingeladen.” – “No, unfortunately I am not invited.”
– “Es ist ok, ich lade dich ein!” – “It’s ok, I am inviting you!”
– “Es sieht nicht sehr einladend aus.” – “It does not look very inviting.”

Though there is some debate about this (and there may be regional differences), when applicable, the German word “einladen” strongly implies that the invitee will not have to pay, whereas the English “to invite” can be used more casually to ask for someone’s presence or participation (without implicitly making any statement about who will foot the bill):

– “Ich lade dich zum Abendessen ein.” – “I’m taking you out for dinner/treating you to dinner.”