Searching for an algorithm with $\Theta(n2^n)$ time complexity
If you just want any algorithm with running time $\Theta(n 2^n)$ you can trivially compute $\ell = n 2^n$ from the input size $n$ and then have a loop of no-ops (or any constant number of &quot;useless&quot; elementary operations) iterate $\ell$ times.
If you are looking for a &quot;reasonable&quot; algorithm for a &quot;well-known&quot; problem with that time complexity, you can look for example at the brute-force solution for the subset-sum problem.
Given an input collection of $n$ non-negative integers $X=\{x_1, x_2, \dots, x_n\}$ and $T \in \mathbb{N}$ the problem asks to decide whether there exists a subset $S$ of $X$ for which $\sum_{x_i \in S} x_i = T$.
A trivial brute-force algorithm for this problem looks at $2^n$ possible binary vectors $v_1, v_2, \dots \in \{0,1\}^n$ of length $n$, where each vector $v_i$ represents a subset of $X$. For each $v_i$ we explicitly check, in $\Theta(n)$ time, whether $(x_1, \dots, x_n) \cdot v_i = T$.
After all vectors have been examined, return &quot;yes&quot; if one such vector was found and &quot;no&quot; otherwise.