That is fun! It's interesting, the solution is presented here[1], but I'm pretty sure it's wrong (off by 19) due to not taking into account the possibility for the pawn to move two squares on its first move.
> Note: If you counted the pawn moving forward two squares with its initial move as distinct from its moving two individual squares, then there are 160 paths. Feeling generous, I gave full credit for either approach.
> Note: If you counted the pawn moving forward two squares with its initial move as distinct from its moving two individual squares, then there are 160 paths. Feeling generous, I gave full credit for either approach.