HMMT 2025 U8

A checkerboard is a rectangular grid of cells colored black and white such that the top-left corner is black and no two cells of the same color share an edge. Two checkerboards are distinct if and only if they have a different number of rows or columns. For example, a $20 \times 25$ checkerboard and a $25 \times 20$ checkerboard are considered distinct.

Compute the number of distinct checkerboards that have exactly 41 black cells.