Revision as of 01:45, 15 June 2024 by Admin (Created page with "Prove that if <math>n \ge 0</math>, then <math display="block"> \sum_{k = 0}^n {n \choose k}^2 = {{2n} \choose n}\ . </math> '' Hint'': Write the sum as <math display="block"> \sum_{k = 0}^n {n \choose k}{n \choose {n-k}} </math> and explain why this is a coefficient in the product <math display="block"> (1 + x)^n (1 + x)^n\ . </math> Use this, together with Exercise, to show that <math display="block"> u^{(2)}_{2n} = \frac 1{4^{2n}}{{2n}\ch...")
ABy Admin
Jun 15'24
Exercise
Prove that if [math]n \ge 0[/math], then
[[math]]
\sum_{k = 0}^n {n \choose k}^2 = {{2n} \choose n}\ .
[[/math]]
Hint: Write the sum as
[[math]]
\sum_{k = 0}^n {n \choose k}{n \choose {n-k}}
[[/math]]
and explain why this is a coefficient in the product
[[math]]
(1 + x)^n (1 + x)^n\ .
[[/math]]
Use this, together with Exercise, to show that
[[math]]
u^{(2)}_{2n} = \frac 1{4^{2n}}{{2n}\choose n}\sum_{k = 0}^n {n \choose k}^2 =
\frac 1 {4^{2n}} {{2n}\choose n}^2\ .
[[/math]]