Generalize an equation for $y_i$ given the following sequence of functions:
$$ \begin{align} y_1 &= \beta_0 + \beta_1x_1 \\ y_2 &= \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2 \\ y_3 &= \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2 + \beta_4x_3 + \beta_5x_1x_3 + \beta_6x_2x_3 + \beta_7x_1x_2x_3 \\ &\vdots \\ y_n &= \textrm{ ? } \\ &\vdots \\ \end{align} $$
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In my efforts to find a solution, I noticed that if you ignore the beta coefficients, then the remaining variables can be written as such:
$$ y_i^* = \prod_i(x_i + 1) = 1 + x_1 + x_2 + x_1x_2 + x_3 + x_1x_3 + x_2x_3 + x_1x_2x_3 + \dots $$
Also, consider the case of $i = 3$ (for example). We see that...
$$ \begin{align} y_3 & = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2 + \beta_4x_3 + \beta_5x_1x_3 + \beta_6x_2x_3 + \beta_7x_1x_2x_3 \\ & = \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \\ \beta_5 \\ \beta_6 \\ \beta_7 \\ \end{bmatrix} \begin{bmatrix} 1 & x_1 & x_2 & x_1x_2 & x_3 & x_1x_3 & x_2x_3 & x_1x_2x_3 \end{bmatrix} \\ \end{align} $$
And...
$$ y_3^* = \prod_{i=1}^{3}(x_i + 1) = \tau_0 + \tau_1 + \tau_2 + \tau_3 + \tau_4 + \tau_5 + \tau_6 + \tau_7 \textrm{, where...} $$ $$ \begin{align} & \tau_0 = 1 \\ & \tau_1 = x_1 \\ & \tau_2 = x_2 \\ & \tau_3 = x_1x_2 \\ & \tau_4 = x_3 \\ & \tau_5 = x_1x_3 \\ & \tau_6 = x_2x_3 \\ & \tau_7 = x_1x_2x_3 \\ & \end{align} $$
So, if there is a way to extract the respective $\tau_i$ terms from $y_i^*$, then we could write $y_i$ as...
$$ y_i = \sum_i\beta_i\tau_i $$
Therefore, generalizing an equation for $\tau_i$ would trivially help generalize an equation for $y_i$. However, it's not clear to me how to go about extracting $\tau_i$ from $y_i^*$.
This may (or may not) be a step the right direction. Thank you for your help in advanced.
One formula is $\displaystyle\sum_{S\subseteq\{1, 2, \ldots n\}}\beta_{\sum_{i\in S}2^{i-1}}\prod_{i\in S}x_i$. The index of $\beta$ is just the sum of $2^{i-1}$ for $i$ being the subscripts of the $x$'s in the term.