I am interested in the continued fraction $$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$

OEIS A104977 states without proof that ${T_k} = {( - 1)^{b(k + 2) + 1}}$ (see A090678) if $b(n)$ denotes the number of non-squashing partitions of $n$ into distinct parts (see A088567).

I obtained the sequence ${T_k}$ with another method which has nothing to do with partitions, but I would be interested to learn why there is a connection with those partitions. Can anyone help me?

**Edit**: Let me give some more information about the sequence $T_n.$

Let $$D(n) = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$$ be a Hankel determinant of the sequence $(a_n), $ where $a_n=1$ if $n+2$ is a power of $2$ and $a_n=0$ else.

Then $$ T_n=D(n)D(n+2).$$ These numbers satisfy $T_{2n}=T_{2n-1}T_{n-1}$ and $T_{2n+1}=-T_{2n}$with initial values $T_0=1$ and $T_1=-1.$

The non-squashing partitions with distinct parts $b(n)$ satisfy $b(2n)=b(2n-1)+b(n)-1$ and $b(2n+1)=b(2n)+1.$

Thus a posteriori it can be observed that $${T_k} = {( - 1)^{b(k + 2) + 1}}.$$

**Question**: Is this a happy coincidence or is there really a connection between the Hankel determinants and the partitions.