Help rendering a complicated sum/product formula

Perhaps you could start with this:

\documentclass{article}
\usepackage{amsmath,relsize}

\begin{document}

\begin{align*}
    \mathlarger{\mathlarger{\mathlarger\sum}}_
        {\substack{B\subset\{1..n\},\\\lvert B\rvert=d}} 
        \Bigl(\prod_
            {\substack{p,q\in B,\\p\neq q}}
            (\alpha_p-\alpha_q)\Bigr)
        \Bigl(\kern-2ex\prod_
            {t\in\{1..n\}\slash B}\kern-2ex
            (x-\alpha_t)\Bigr)
\end{align*}

\end{document}

I tried to replicate it.

The video https://youtu.be/LFrdqQZ8FFc might also be helpful.

I would type it in a slightly different way, to improve the legibility of the formula:

\documentclass{article}
\usepackage{mathtools, relsize}

\begin{document}

\[ \mathlarger{ \sum}_{\substack{B\subset\{1 \cdot\cdot n\}\\ |B| = d}}\,\Bigl(\smashoperator{\prod_{\substack{p, q\in B \\ p\ne q}}}(\alpha_p-\alpha_q)\:\smashoperator{ \prod_{\substack{t \in\{1 \cdot\cdot n\}\\ t \notin B }}}(x-\alpha_t) \Bigr) \]%

\end{document} 

I'd use standard sizes:

\documentclass{article}
\usepackage{amsmath,mathtools}

\newcommand{\tprod}{\mathop{\mathchoice{\textstyle\prod}{\prod}{\prod}{\prod}}}
\newcommand{\sss}{\scriptscriptstyle}

\begin{document}

\begin{equation*}
\sum_{\substack{\sss B\subset\{1..n\}, \\ \sss \lvert B\rvert=d}}\;
  \bigl(
    \smashoperator{\tprod_{\substack{\sss p,q\in B, \\ \sss p\neq q}}}
    (\alpha_p-\alpha_q)
  \bigr)
  \bigl(
    \smashoperator{\tprod_{\sss t\in\{1..n\}/B}}
    (x-\alpha_t)
  \bigr)
\end{equation*}

\end{document}