The Sum of Binomial Coefficients with Even Indices

In combinatorics, operations involving binomial coefficients can often reveal elegant and surprising identities. One such problem involves the sum of binomial coefficients where the indices are even, as described in the expression provided in the initial content. This article aims to explore this problem in detail, providing a deeper understanding of the underlying combinatorial identities and their significance.

Introduction

Consider the expression:

C_n^0 cdot C_{0}^0 C_n^{n/2} cdot C_2^1 C_n^{n-2/(2-2)} cdot C_{2-2}^{(n-2-2)/2} ldots C_n^{2i} cdot C_{2i}^i C_{n-2i}^{(n-2i)/2} ldots C_n^n cdot C_n^{n/2} cdot C_0^0, where n 2m

Understanding the Expression

The expression involves binomial coefficients where the indices are even. The binomial coefficient C_n^k is defined as the number of ways to choose k elements from a set of n elements, which can be expressed as:

C_n^k frac{n!}{k!(n-k)!}

Key Observations and Simplifications

By applying the binomial coefficient definition, we can simplify the expression. For the i-th term in the given summation, we find:

binom{2m}{2i} cdot binom{2i}{i} cdot binom{2m-2i}{m-i} binom{2m}{m} cdot binom{m}{i}^2

This simplification allows us to rewrite the entire summation as:

binom{2m}{m} sum_{i0}^{m} binom{m}{i}^2

Proving the Identity

To resolve the summation, we need to prove the identity:

sum_{i0}^{m} binom{m}{i}^2 binom{2m}{m}

Proving this identity can be done using mathematical induction, which is a method of proving statements that follow a specific pattern. The base case (m0 and m1) can be verified directly. Assuming the identity holds for mk, we need to show it holds for mk 1:

Base Case

For m0 and m1, the identity can be verified manually:

For m0, binom{0}{0}^2 1 and binom{0}{0} 1, so the identity holds. For m1, binom{1}{0}^2 binom{1}{1}^2 1 1 2 and binom{2}{1} 2, so the identity holds.

Inductive Step

Assuming the identity holds for mk, we need to show it holds for mk 1:

sum_{i0}^{k 1} binom{k 1}{i}^2 binom{2k 2}{k 1}

Using the binomial identity:

binom{2(k 1)}{k 1} sum_{i0}^{k 1} binom{k 1}{i} cdot binom{k 1}{k 1-i}

This can be expanded as:

binom{2(k 1)}{k 1} sum_{i0}^{k 1} binom{k 1}{i}^2 sum_{i0}^{k} binom{k 1}{i} cdot binom{k 1}{k 1-i}

Using the induction hypothesis:

sum_{i0}^{k} binom{k 1}{i} cdot binom{k 1}{k 1-i} sum_{i0}^{k} binom{k}{i} cdot binom{k}{k-i} binom{2k}{k} binom{2k}{k 1}

Thus, the identity holds for mk 1.

Conclusion and Significance

The final outcome of the summation, given that n 2m, is:

(binom{2m}{m})^2

This result highlights the elegance and beauty of combinatorial identities and their applications in solving complex problems. The sum of binomial coefficients with even indices not only provides insights into the structure of combinatorial objects but also demonstrates the powerful techniques available in algebraic combinatorics.

Keywords

binomial coefficients even indices combinatorial identities

References

For those interested in further exploration of combinatorics and binomial coefficients, the following references provide deeper insights:

Richard Stanley (2015), Enumerative Combinatorics, Volume 2 Knuth, D. E. (1997), The Art of Computer Programming, Volume 4 Grimaldi, R. P. (2004), Discrete and Combinatorial Mathematics: An Applied Introduction