Elementary-symmetric expansion of a specific plethysm

Determine the expansion of the plethysms s_{1^i}[s_{1,1}]=e_i[F_2] in terms of elementary symmetric functions for all integers i≥1.

Background

The Schur expansion of s_{1^i}[s_{1,1}] is known from work of Littlewood and subsequent authors. However, the paper distinguishes this from the expansion in the elementary symmetric-function basis, which is the form needed for the associated Reidemeister-spectrum computations. The general elementary-symmetric expansion remains unresolved even in this comparatively structured case.