S_{2} = sum k = 0 to n k * binomial(n,k) ^ 2

The sum you provided can be expressed as:

\[
S_2 = \sum_{k=0}^{n} k \cdot \binom{n}{k}^2
\]

This is a sum that combines binomial coefficients, and finding a closed form for such sums often involves combinatorial identities or generating functions.

### Step-by-Step Approach

1. **Understanding the General Form:**
   The binomial coefficient \(\binom{n}{k}\) counts the number of ways to choose \(k\) objects from \(n\) distinct objects. The square of the binomial coefficient, \(\binom{n}{k}^2\), appears in many combinatorics problems, especially those involving counting lattice paths.

2. **Expanding the Sum:**
   The given sum can be rewritten as:

   \[
   S_2 = \sum_{k=0}^{n} k \cdot \binom{n}{k}^2
   \]

3. **Using Symmetry Properties:**
   One helpful approach to evaluating sums involving binomial coefficients is through known identities such as the Chu-Vandermonde identity or using properties of generating functions.

4. **Known Result:**
   The sum of the form \(\sum_{k=0}^{n} k \cdot \binom{n}{k}^2\) has been studied in combinatorics, and the closed form for this sum is:

   \[
   S_2 = n \cdot \binom{2n-1}{n-1}
   \]

### Final Expression:
Thus, the closed-form expression for \( S_2 \) is:

\[
S_2 = n \cdot \binom{2n-1}{n-1}
\]

This is the result of summing \( k \cdot \binom{n}{k}^2 \) over \( k \) from \( 0 \) to \( n \).

Would you like a more detailed derivation or explanation of the steps?

### Related Questions:
1. What is the Chu-Vandermonde identity and how is it used in binomial sums?
2. How do generating functions help in summing binomial coefficient-related expressions?
3. Can this method be extended to higher powers of \( k \) in similar sums?
4. How do lattice paths relate to binomial coefficient squared sums?
5. What are other known sums involving \( \binom{n}{k}^2 \)?

### Tip:
When dealing with sums involving binomial coefficients, exploring symmetry and known combinatorial identities can simplify the process.

S_{2} = sum k = 0 to n k * binomial(n,k)^2

Explore the summation of k * binomial(n,k)^2 with a detailed solution. This combinatorics problem utilizes binomial coefficient properties and yields a closed form using the Chu-Vandermonde identity. Suitable for undergraduate math students.

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