Comptine /kɔ̃tin/ is the French for nursery rhyme or for a counting rhyme or song. I learnt it last night and thought I’d look into where it comes from.
According Wiktionnaire, comptine is made up of compte (count, number, account) and the suffix -ine. Compte /kɔ̃t/ comes from computus (count, number, account, calculation), from computo (to count – computer in French), from con- (suffix = with; all) and putō (to think, suppose, reckon, count, prune), perhaps from the Proto-Indo-European *pu- (to wash).
My French dictionary says that comptine refers particularly to nursery rhymes involving counting, which is reflected in its etymology. Other words for children’s songs include chansons pour enfant (songs for children) and berceuse (lullaby, cradle song, hushaby, rocking chair), which also means nursemaid, and comes from bercer (to rock, cradle, lull), which apparently comes from Gaulish.