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 of "partition Eisenstein series'', extensions of the classical Eisenstein series $E_{ 2k}(q)$ defined by $$\lambda=(1^{ m_1}, 2^{ m_2},\dots, n^{ m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E_{ \lambda}(q):= E_2(q)^{ m_1} E_4(q)^{ m_2}\cdots E_{ 2n}(q)^{ m_n}. $$ For functions $\phi : \mathcal{ P}\mapsto \mathbb{ C}$ on partitions, the weight $2n$ partition Eisenstein trace is $$ \text{ Tr}_n(\phi;q):=\sum_{ \lambda \vdash n} \phi(\lambda)E_{ \lambda}(q). $$ For all $t$, we prove that $U_{ 2t}(q)=\text{ Tr}_t(\phi_U;q)$ and $V_{ 2t}(q)=\text{ Tr}_t(\phi_V;q),$ where $\phi_U$ and $\phi_V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.
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