Problem 44735. Aztec Diamond domino tilings

Consider a Cartesian grid, with verteces at integer x and y values, where every four vertices around a vacant space define a unit square. An Aztec Diamond of order d is the shape formed by all unit squares whose centers satisfy the equation:

abs(x) + abs(y) <= d

Given the order of an Aztec Diamond, d (positive integer), return the number, n, of possible tilings using domino tiles, i.e. rectangles sized 1x2 and 2x1, such that:

1. The entire shape is covered
2. There are no overlapping tiles
3. None of the tiles stick out of the shape

Example:

An Aztec Diamond of order 4 is shown at this URL.

Input: d = 4

Output: n = 1024