On a two-color partition series and its companions

We study the two-color distinct-part series $S_1(q)$, equivalently Andrews’ generating function $v_d(q)$ for strictly concave compositions, and its odd and even companions $T_o(q)$ and $T_e(q)$. We determine the coefficients of $S_1(q)$ modulo $4$ and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion $C(q)=(q;q{)}_{\mathrm{\infty }}{T}_o(q)$, we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that $c(n)$ can be nonzero only when $24n+28$ is represented by $x^2+3y^2$.