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There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and $$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for $n\geq1$}.$$ What I have found is the following multi-variable polynomial generalization. Let $G_{-1}=0, G_0=x_0$ and $$G_n=x_nG_{n-1}+G_{n-2} \qquad \text{for $n\geq1$}.$$ It is possible to write down $G_n$ explicitly $$G_n=\sum_{\substack{k=1 \\ k+n\equiv1\,(mod\, 2)}}^n\sum_{\substack{0\leq i_1<\cdots<i_k\leq n \\ i_1+\cdots+i_k+k\equiv 1\,(mod\, 2)}}x_{i_1}x_{i_2}\cdots x_{i_k}.$$

QUESTION. Is $G_n$ known in the literature? Is there a generating function for $G_n$?

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  • $\begingroup$ Have you tried looking at the linear transformation version? That is, the linear map $(G_{n-2},G_{n-1}) \mapsto (G_{n-1},G_n)$ can be represented as a $2\times 2$ matrix. This may be related to continued fractions. That is, the map $G_{n-1}/G_{n-2} \mapsto G_n/G_{n-1}$. $\endgroup$
    – Somos
    Commented Jul 11, 2022 at 1:38
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    $\begingroup$ I believe you are looking for the continuant, see en.wikipedia.org/wiki/Continuant_(mathematics) $\endgroup$
    – Gerry Myerson
    Commented Jul 11, 2022 at 4:01
  • $\begingroup$ Thank you, both. $\endgroup$
    – T. Amdeberhan
    Commented Jul 11, 2022 at 15:43

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