Residue Restrictions for a Two-Color Partition Series

We study residue restrictions for a two-color partition series $P(q)=(q^4;q^4{)}_{\mathrm{\infty }}S(q)$ arising from work of Andrews and Bachraoui on partitions with odd smallest part. Motivated by the explicit exponent structure in the Bailey-transform formulas for the associated generating function, we obtain elementary restrictions on the support of the coefficients. In particular, we show that the residue class $4(\mathrm{m}\mathrm{o}\mathrm{d}8)$ does not occur, and we prove a further refinement modulo $16$, yielding several vanishing classes for the coefficients of the series. Our argument is completely residue-theoretic and avoids the use of modular completions.