  In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse
sequence, and proved that all the Hankel determinants of the period-doubling
sequence are odd integral numbers. We speak of $t$-extension when the entries
along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we
prove that the $t$-extension of each Hankel determinant of the period-doubling
sequence is a polynomial in $t$, whose leading coefficient is the {\it only
one} to be an odd integral number. Our proof makes use of the combinatorial
set-up developed by Bugeaud and Han, which appears to be very suitable for this
study, as the parameter $t$ counts the number of fixed points of a permutation.
Finally, we prove that all the $t$-extensions of the Hankel determinants of the
regular paperfolding sequence are polynomials in $t$ of degree less than or
equal to $3$.
