Question

If `(1+x)^n = c_0 + c_1x + c_2x^2 + ... + c_nx^n` then show that `c_0c_1 + c_1c_2 + ... + c_(n-1)c_n = (2n)!(n+1)!(n-1)!`?

Answer 1

The relationship is invalid

Explanation:

We can readily show that this relationship is invalid by using a counter example:

Consider the case `n=2`:

`(1+x)^2 = 1 + 2x + x^2`

Then:

`LHS = c_0c_1 + c_1c_2`
`\ \ \ \ \ \ \ \ = 1.2 + 2.1`
`\ \ \ \ \ \ \ \ = 2 + 2`
`\ \ \ \ \ \ \ \ = 4`

`RHS = (2n)!(n+1)!(n-1)!`
`\ \ \ \ \ \ \ \ = 4!3!1!`
`\ \ \ \ \ \ \ \ = 24.6.1`
`\ \ \ \ \ \ \ \ = 144`

Hence the relationship is false.