Bishop routes on the diagonals

A bishop moves on diagonals, so rotate your view: turn the board $45$ degrees and the diagonals become rows and columns. Counting diagonal zigzag routes then becomes the same additive counting as before, and Pascal's triangle appears along the diagonals. Each square's route count is the sum of the counts of the squares that reach it in one diagonal step.

Because a bishop stays on one colour, only half the squares are reachable, and the route counts you build are again binomial coefficients, read along the diagonal direction instead of the horizontal.