Let Sn denote the sum of the first n natural numbers?

Question

let Sn denote the sum of the first n natural numbers. prove that

2 (s1*s2n + s2*s(2n-1) + s3*s(2n-2) + ... + sn*s(n+1) ) = (2n+5)! / 4! (2n-1)!.

Answer 1

Clearly not true, so it can't be proven

Let n = 1

then Sn = S1 = 1, S2n = S2 = 2

there is only one term on the left hand side so the left side =

2(1*2) = 4

On the right, we have 2n+5 = 7 and 2n-1 = 1 so:

(2n+5)! /(4! (2n-1)!) = 7! / (4 ! 1!) = 5 * 6 * 7 = 210

4 does not equal 210.