IMC 2018 Problem 8

Let $\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}$ . A frog moves along the points of $\Omega$ by jumps of length $1$ . For every positive integer $n$ , determine the number of paths the frog can take to reach $(n,n,n)$ starting from $(0,0,0)$ in exactly $3n$ jumps.

Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University