  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.
