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We prove that the class of permutations generated by passing an orderedsequence $12\dots n$ through a stack of depth 2 and an infinite stack in seriesis in bijection with an unambiguous context-free language, where a permutationof length $n$ is encoded by a string of length $3n$. It follows that thesequence counting the number of permutations of each length has an algebraicgenerating function. We use the explicit context-free language to compute thegenerating function: \begin{align*} \sum_{n\geq 0} c_n t^n &=\frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q}\end{align*} where $c_n$ is the number of permutations of length $n$ that canbe generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simplevariant of the Catalan generating function. This in turn implies that$c_n^{1/n} \to 2+2\sqrt{5}$.

Permutations generated by a depth 2 and infinite stack in series are algebraic