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When dealing with binomial coefficients, especially in problems involving the binomial theorem, it's important to understand their notation and calculation. The binomial coefficient \(\binom{k}{n}\) is defined as \[ \binom{k}{n} = \frac{k(k-1)(k-2)...(k-(n-1))}{n!} \]. This makes it easy to calculate coefficients in polynomials or series expansions in combinatorial contexts. For instance, in the given exercise, we are using \(\binom{-1/2}{n}\), which follows the same structure but with specific values for \(k\) and \(n\). It's necessary to simplify the expression by utilizing the properties of factorials and fractions. By acknowledging these steps, you can decipher complex expressions and match them appropriately, such as transforming double factorial into binomial coefficients.

The extended binomial theorem is a generalization of the conventional binomial theorem. It allows calculating the expansion of expressions involving negative or fractional exponents. The standard binomial theorem expands \( (1 + x)^{n} \) as \[ \sum_{k=0}^{n} \binom{n}{k} x^{k} \], but the extended version applies to \( (1 + x)^{\alpha} \), where \(\alpha\) can be any real number, not just integers. In the exercise, for \( (1 - 4x)^{-1/2}\), use the series form \[ (1 + x)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n} \]. This is crucial for expanding more complicated expressions and determining coefficients of particular terms like \(x^{n}\). Understanding this theorem simplifies many algebraic and combinatorial problems.

Combinatorial identities form the backbone of many algebraic manipulations in mathematics. These identities relate different binomial coefficients and make solving equations easier. One notable identity seen in the problem is \[ \binom{2n}{n} = \frac{(2n)!}{n!n!} \]. This showcases how factorials are essential in deriving binomial expressions. Additionally, recognizing double factorials and simplifying them \( (2n-1)!! = \frac{(2n)!}{2^{n}n!} \), helps transform complex binomials into manageable forms. Utilizing these identities ensures precision and correctness when dealing with polynomials, series expansions, and high combinatorics.

Series expansion involves expressing a function as an infinite sum of terms calculated from the function's derivatives at a single point. The primary benefit lies in approximating complicated functions with simpler polynomial forms. For example, in the given problem, the series expansion of \( (1 - 4x)^{-1/2} \) is \[ \sum_{n=0}^{\infty} \binom{-1/2}{n} (-4x)^{n} \], which we then simplified. By substituting and simplifying, the coefficient of \( x^{n} \) in the expanded form is \( \binom{2n}{n} \). This method dramatically simplifies how we handle and understand functions in mathematics, providing clear paths for evaluating expressions and coefficients analytically. Always break down complicated terms systematically, simplify by substitutions, and use known series expansions if possible.