Derivatives of theta functions as Traces of Partition Eisenstein series

In his ``lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of q-series ₂ₜ(q)\ and ₂ₜ(q)\ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of ``partition Eisenstein series'', extensions of the classical Eisenstein series E₂ₖ(q) defined by λ=(1m₁, 2m₂,, nmₙ) n \ \ \ \ \ \ \ \ \ \ Eλ(q):= E₂(q)m₁ E₄(q)m₂⋯ E₂ₙ(q)mₙ. For functions φ : P↦ on partitions, the { weight $2n$ partition Eisenstein trace} is ₙ(φ;q):=∑λ n φ(λ)Eλ(q). For all t, we prove that U₂ₜ(q)=ₜ(φᵤ;q) and V₂ₜ(q)=ₜ(φᵥ;q), where φᵤ and φᵥ are natural partition weights, giving the first explicit quasimodular formulas for these series.