Special Case of Vandermonde's Identity

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This Demonstration shows a special case of Vandermonde's identity: the sum of the squares of the elements in the middle row of the diamond equals the element at the bottom of the diamond. The diamond is a condensed form of Pascal's triangle. The "number of rows" slider corresponds to the number of rows displayed in the Pascal diamond, while the "path index" slider represents a possible path to get from the top element to the bottom element of the triangle, going either diagonally down left or down right each move.

Contributed by: Kevin Ren and Ananth Rao (March 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Vandermonde's identity states that for all positive integers , , , the following identity holds [1]:

.

In the special case of , the identity reduces to

.

It is a well-known fact of Pascal's triangle that the entry of the row is (where indexing for rows and entries starts from 0), and the number at the bottom of the diamond is . Thus, it is sufficient to show that the sum of the squares of the elements in the middle row is equal to the number at the bottom of the diamond.

It is also well-known that the number of paths from the top element to another element, traveling only in the steps down left or down right, is equal to the value of the element. Thus, the number of red paths from the top of the diamond to the bottom is , and the number of paths from the top element to entry in the middle row is . By symmetry, there are also paths from row , entry to the bottom element of the diamond, so there are paths from the top element of the diamond to the bottom passing through element in the middle row. Each path from the top element to the bottom element must pass through exactly one element in the middle row, so that

.

Snapshot 1: Pascal's triangle for 18 rows; there are actually 19 rows, since row indices start at 0

Snapshot 2: Pascal's triangle for four rows and a path passing through the 2 in the middle row

Snapshot 3: Pascal's triangle for 14 rows and a path passing through a 35 in the middle row

This was a project for Advanced Topics in Mathematics II, 2017–2018, Torrey Pines High School, San Diego, CA.

Reference

[1] E. W. Weisstein. "Chu–Vandermonde Identity" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/Chu-VandermondeIdentity.html (Wolfram MathWorld).



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