An archaic auxiliary (historically written for 'む') expressing probability or conjecture ('probably, ...-will'), used in classical or literary Japanese.
明日は晴れん、と古い日記に記してある。
The old diary says 'It will probably be sunny tomorrow.'
彼は来られんと信じていた。
He believed that he probably would not be able to come.
An archaic auxiliary expressing volition or intent ('will; intend to'), corresponding to classical 'む'.
我行かん、という覚悟を固めた（古文的表現）。
I resolved 'I will go' (classical-style expression).
若者は新しい道を拓かんとする気概があった。
The young man had the spirit to open up a new path (intended/willed to).
An archaic use expressing a wish that someone would do something for the speaker (roughly 'would like to have done [for me]'). Rare in modern Japanese.
誰か助けてくれんものかと祈るのみだった（古文調）。
All I could do was pray that someone would help me (archaic tone).
あの時、来てくれんであったらと語る古老の話が残る。
There remains an old tale saying 'If only he had come then' (archaic phrasing).