Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this, we establish a connection between nested words and series-parallel-biposets. Applying our result, we obtain a characterization of algebraic formal power series in terms of weighted logics. This generalizes a result of Lautemann, Schwentick and Therien on context-free languages.
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