Periodic vanishings of the Legendre-17 signed partition numbers

For $f:\mathbb{N}\to \{0,\pm 1\}$ the $f$-signed partition numbers $\mathfrak{p}(n,f)$ are defined to be the weighted partition sums $$\mathfrak{p}(n,f)=\sum _{\begin{array}{c}{x}_1+\cdots +x_k=n\\ x_1\ge \cdots \ge x_k>0\\ k\ge 1\end{array}}f(x_1)f(x_2)\cdots f(x_k)\mathrm{.}$$ For prime $p>2$, let $(\frac{\cdot }{p})$ denote the Legendre symbol modulo $p$. The first half of this paper derives Rademacher-style series formulae for the quantities $\mathfrak{p}(n,\pm (\frac{\cdot }{p}))$ for $p<24$ satisfying $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ (that is, for $p=5,13,17$), and the extensions to general $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ are made apparent in our derivations. In the second half of this paper, the series formulae for $\mathfrak{p}(n,\pm (\frac{\cdot }{17}))$, as well as various properties of Dedekind sums and their “character-twisted” analogues, are used to establish that these two quantities are identically zero on certain (mod $34$)-arithmetic progressions.