We introduce the notion of su(2) spin-s Dicke states, which are higher-spin generalizations of usual (spin-1/2) Dicke states. These multi-qudit states can be expressed as superpositions of su(2s+1) qudit Dicke states. These states satisfy a recursion formula, which we use to formulate an efficient circuit for their preparation. The algorithm is deterministic and does not require ancillary qudits.