  In 1998, Allouche, Peyri\`ere, Wen and Wen established that the Hankel
determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always
nonzero. Their proof depends on a set of sixteen recurrence relations. We
present an alternative, purely combinatorial proof of the same result. We also
re-prove a recent result of Coons on the non-vanishing of the Hankel
determinants associated to two other classical integer sequences.
