BALTIC-WAY 2025 Problem 8

A sequence of non-negative integers $a_1, \ldots, a_n$ is called self-descriptive if

$$a_k = |\{i \mid a_i \geq k\}|$$

for each $k \in \{1, \ldots, n\}$, i.e. $a_k$ equals the number of elements of the sequence that are greater than or equal to $k$, for each $k \in \{1, \ldots, n\}$.

For each positive integer $n$, determine the number of self-descriptive sequences of length $n$.