What does the formal definition of "closed-form" say about finite sums exactly?

Question

I have looked through the online literature and there seems to be conflicting answers to this question. Consider the finite sum

$$\sum_{i=0}^{\lfloor n/2 \rfloor} {n-1\choose i}$$

Is this expression considered closed-form?

These two links say that such a finite sum is not considered closed-form:

1. https://opendsa-server.cs.vt.edu/ODSA/Books/CS3/html/Summations.html#:~:text=This%20is%20known%20as%20a%20closed%2Dform%20solution
2. http://www3.govst.edu/wrudloff/CPSC438/CPSC438/CH05/Chapter5/Section.5.2.pdf

while this link says that such a finite sum is considered closed-form:

3. https://en.wikipedia.org/wiki/Closed-form_expression#:~:text=Yes-,Finite%20sum,Yes,-Finite%20product

As far as I know, a closed-form expression has a finite number of standard operations or known functions. Based on this, my hunch is that the sum above is closed-form.

Answer 1

Hint: The notion of closed form in mathematics is context dependent. Finite sums of type \begin{align*} \sum_{i=0}^{\lfloor n/2 \rfloor} {n-1\choose i} \end{align*} are usually not considered to be in closed form, since there is bound index variable $i$.

We find in Chapter I: What Is Enumerative Combinatorics? in the classic Enumerative Combinatorics, Vol. I by R. P. Stanley:

The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually we are given an infinite collection of finite sets $S_i$ where $i$ ranges over some index set $I$ (such as the nonnegative integers $\mathbb{N}$), and we wish to count the number $f(i)$ of elements in each $S_i$ simultaneously. Immediate philosophical difficulties arise. What does it mean to count the number of elements of $S_i$? There is no definite answer to this question. Only through experience does one develop an idea of what is meant by a determination of a counting function $f(i)$. The counting function $f(i)$ can be given in several standard ways:

• The most satisfactory form of $f(i)$ is a complete explicit closed formula involving only well-known functions, and free from summation symbols.

Note: The crucial aspect is the term bound variable. As long as there are bound variables with limited scope given by $\Sigma$ or $\prod$ symbols, the expression is not considered to be in closed form.