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 $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms ofpartition Eisenstein series'', extensions of the classical Eisenstein series $E{2k}(q)$ defined by $$\lambda=(1^{m1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E{\lambda}(q):= E2(q)^{m_1} E4(q)^{m2}\cdots E{2n}(q)^{mn}. $$ For functions $\phi : \mathcal{P}\mapsto \C$ on partitions, the {\it weight $2n$ partition Eisenstein trace} is $$ \Trn(\phi;q):=\sum{\lambda \vdash n} \phi(\lambda)E{\lambda}(q). $$ For all $t$, we prove that $U{2t}(q)=\Trt(\phiu;q)$ and $V{2t}(q)=\Trt(\phiv;q),$ where $\phiu$ and $\phi_v$ are natural partition weights, giving the first explicit quasimodular formulas for these series.