Nemotron-IMO-Bench 2026 C8

Let $\sigma(x)$ be the sum of positive divisors of the positive integer $x$. Prove that for all integers $n>1$ $$\sum_{k=0}^{n-1} (-1)^k (2k+1)\sigma (\frac{n^2+n}{2}-\frac{k^2+k}{2})=(-1)^{n-1}\frac{n(n+1)(2n+1)}{6}.$$