What mathematical object accepts a sequence as input? A function inputs a single number and outputs a single number. (e.g. $y = f(x)$)
What mathematical object inputs a sequence, or even just a vector or tuple?
Edit:
I understand that a function doesn't have to take a single number and output a single number.  But what do you call the functions that don't?  Can you give me some Google-able terms, please?
Example #1: the "derivative" (calculus) operation inputs a function and outputs a function.  So it is clearly different than the basic/classic/algebraic "function" $y = f(x)$.
Example #2: the "union" (set-theory) operation inputs two sets and outputs a set.
Can you provide an example of any operation that inputs a sequence?

Well, actually... the real numbers are equivalent to rational sequence classes which converge to that  number so any real number function is equivalent to a function taking in a sequence as an input. (Although the domain the domain of one representative sequence from an infinite class of sequences, all converging to the same point might be whiffy.)
" So it is clearly different than the basic/classic/algebraic "function" y=f(x)"
Actually it's not as clear as you'd think.  The only real (pun unintended) difference is those algebraic functions have a specific domain and range.  So for as I know there aren't any terms to differentiate functions that use numbers vs. anything else.  If anything, it's real valued functions that are the exception and distinguished by the term "real valued functions".
So for googling terms I don't have many suggestions other than "functions that take sequences as input".
Examples of functions with sequences are plentiful.  F: sequence space -> extended reals, F({a_n} = lim {a_n} is obvious.  
