In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two families of zeta functions of Arakawa-Kaneko-type by means of inverse binomial series involving harmonic sums which appeared fifteen years ago in physics in relation with the Feynman diagrams. In certain cases, these series may be explicitly evaluated in terms of zeta values and other related numbers. Incidentally, this connection allows us to deduce new identities for the constant $C= \sum_{n\geq 1} \frac{1}{(2n)^3}(1+\frac13 + \dots + \frac{1}{2n-1})$ considered by S. Ramanujan in his notebooks.