IMC 2016 Problem 9

Let $k$ be a positive integer. For each nonnegative integer $n$ , let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$ . Prove that for every $n\ge1$ , we have $f(n-1)f(n+1)\leq f(n)^2$ .

(Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro)