  We prove the equality of several $\tau$-recurrent sequences, which were first
considered by Pellarin, and which have close connections to Drinfeld vectorial
modular forms. Our result has several consequences: an $A$-expansion for the
$l^\text{th}$ power ($1 \leq l \leq q$) of the deformation of the weight 2
Eisenstein series; relations between Drinfeld modular forms with
$A$-expansions; a new proof of relations between special values of Pellarin
$L$-series.
