Problem 44072. Number of paths on a n-dimensional grid

This problem is inspired by https://www.mathworks.com/matlabcentral/cody/problems/1483-number-of-paths-on-a-grid and https://www.mathworks.com/matlabcentral/cody/problems/44066-number-of-paths-on-a-3d-grid, which you might want to solve first.

Consider n-dimensional grid, and you are moving from one corner to the farthest corner in a minimal number of moves. Each move corresponds to moving to a neighbouring hypercube (among possible up to 2*n neighbours). How many ways are there?

Input format is a row array of size "d" (for d dimension) with number of grid points on each direction.

Optional: can you solve it without loops?