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The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the determinant of the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.

Remark. For reference, here are the sequences of Hankel determinants ($n\geq1$) associated with $\{{\color{blue}a_k}\}_{k\geq0}, \{{\color{blue}b_k}\}_{k\geq0}, \{{\color{blue}c_k}\}_{k\geq0}$, respectively, \begin{align*} &1, -2, 4, 8, -16, -32, -64, 128, -256, -1536, \dots \\ &1,-2,-4,8,16,-32,-64,128,4864,-9728,\dots \\ &1,-2,-4,8,48,-96,-320,640,1792,512,\dots \end{align*}

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the determinant of the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.

Remark. For reference, here are the sequences of Hankel determinants ($n\geq1$) associated with $\{{\color{blue}a_k}\}_{k\geq0}, \{{\color{blue}b_k}\}_{k\geq0}, \{{\color{blue}c_k}\}_{k\geq0}$, respectively, \begin{align*} &+1, -2, +4, +8, -16, -32, -64, +128, -256, -1536, \dots \\ &+1,-2,-4,+8,+16,-32,-64,+128,+4864,-9728,\dots \\ &+1,-2,-4,+8,+48,-96,-320,+640,+1792,+512,\dots \end{align*}

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the determinant of the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the determinant of the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.

Remark. For reference, here are the sequences of Hankel determinants ($n\geq1$) associated with $\{{\color{blue}a_k}\}_{k\geq0}, \{{\color{blue}b_k}\}_{k\geq0}, \{{\color{blue}c_k}\}_{k\geq0}$, respectively, \begin{align*} &1, -2, 4, 8, -16, -32, -64, 128, -256, -1536, \dots \\ &1,-2,-4,8,16,-32,-64,128,4864,-9728,\dots \\ &1,-2,-4,8,48,-96,-320,640,1792,512,\dots \end{align*}

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.

The present discussion finds its motivation in the comments by Ira Gessel to my earlier MO question. More specifically,
$$\prod_{i\geq0}(1-x^{2^i})=\sum_{k\geq0}(-1)^{s_2(k)}x^k$$ where $s_2(k)$ is the sum of the binary digits of $k$. Side remark: $\{(-1)^{s_2(k)}\}_{k\geq0}$ is known as the Thue-Morse sequence.

Given a sequence $(a_k)$, the determinant of the matrix $M$ having entries $M_{i,j}=a_{i+j-2}$ (for $1\leq i,j\leq n$) is called its Hankel transform. Several authors studied the Hankel transform of the Thue-Morse sequences noted above, see Hao Fu and Guo-Niu Han and references therein. Fu-Han explored other sequences arising from (I'll only focus on three of them) \begin{align*} \sum_{k\geq0}{\color{blue}a_k}x^k &=\prod_{i\geq0}(1-x^{2^i}), \\ \sum_{k\geq0}{\color{blue}b_k}x^k &=\prod_{i\geq0}(1-x^{3^i}-x^{2\cdot3^i}), \\ \sum_{k\geq0}{\color{blue}c_k}x^k &=\prod_{i\geq0}(1-x^{5^i}-x^{2\cdot5^i}-x^{3\cdot5^i}+x^{4\cdot5^i}). \end{align*} Let $t_{p,r}(k)$ denote number of $r$'s in base-$p$ digital expansion of $k$. By contrast to Fu-Han, I like to define three other sequences $$\tilde{{\color{red}a}}_k=(-1)^{t_{2,1}(k)}, \qquad \tilde{{\color{red}b}}_k=(-1)^{t_{3,2}(k)}, \qquad \tilde{{\color{red}c}}_k=(-1)^{t_{5,2}(k)}.$$

First, observe that $t_{2,1}(k)=s_2(k)$ and $\tilde{a}_k=a_k$. So, the Hankel transforms of the latter are equal. On the other hand, although all sequences are $\pm1$, the pair $\tilde{b}_k$ and $b_k$ as well as the pair $\tilde{c}_k$ and $c_k$ are unequal sequences in general. Take, for instance, \begin{align*} \sum_{k\geq0}{\color{blue}b}_kx^k&=1+x-x^2+x^3+x^4-x^5-x^6-x^7+x^8+x^9+\cdots \\ \sum_{k\geq0}\tilde{{\color{red}b}}_kx^k &=1-x-x^2-x^3+x^4+x^5-x^6+x^7+x^8-x^9+\cdots \end{align*}

QUESTION. Are the Hankel transforms of $\tilde{{\color{red}b}}_k$ and ${\color{blue}b}_k$ equal? Are the Hankel transforms of $\tilde{{\color{red}c}}_k$ and ${\color{blue}c}_k$ are equal? Experiments suggest they are equal.