We study a bosonic string with one end free and the other confined to a D-brane. Only the odd oscillator modes are allowed, which leads to a Virasoro algebra of even Virasoro modes only. The theory is quantized in a gauge where world-sheet time and ordinary time are identified. There are no negative or null norm states, and no tachyon. The Regge slope is twice that of the open string; this can serve as a test of the usefulness of the the model as a semi-quantitative description of mesons with one light and one extremely heavy quark when such higher spin mesons are found. The Virasoro conditions select specific SO(D-1) irreps. The asymptotic density of states can be estimated by adapting the Hardy-Ramanujan analysis to a partition of odd integers; the estimate becomes exact as D goes to infinity
