The standard notation for the set of all subsets (power set) of a set $A$ is either $\wp(A)$ or $2^A$.
I'd like to know whether there's a standard notation and name for the set of all sequences of length from $1$ to $n$ that can be formed from a set $A$, where $n$ is the cardinality of $A$.
That is, where $A = \{a_1, a_2, ..., a_n\}$, I want to know if there's a standard name and notation for the set:
$$\{(a_1), (a_2), ..., (a_1, a_1, a_1), ..., (a_4, a_5, a_1), ..., (a_2, a_1, a_3, a_3), ..., (a_n, ..., a_{n-1}), (a_n, ..., a_n)\},$$
where the last two sequences have $n$ terms.
There is no standard notation for this exact construction that I am aware of, but it can be written rather succinctly still (btw, I'm assuming you mean sequence and not series) as
$$ \bigcup_{i=1}^{n} A^{[i]} $$
If you have not seen this notation before
$$ A^X $$
Denotes the set of functions from $X$ to $A$ (this is motivated by $n^m = |[n]^{[m]}|$ )
And a sequence in $A$ of length $n$ is formally defined as a function from $[n] \to A$