Fitting unbreakable large equation on one line in a twocolumn layout

The cuted package (from the sttools bundle) can temporarily leave two columns mode with its strip environment. An example with the equation entered in the second column:

\documentclass[twocolumn]{ltxdoc}
\usepackage{mathtools, cuted}
\usepackage{lipsum, color}

\begin{document}
\lipsum[1-2]

\lipsum[1]

\textsf{\color{red}Equation entered here: }

\begin{strip}
\begin{equation}
  \sum_{q_\mathrm{total}=0}^{q_A+N_A+q_B+N_B-2}\left(\sum_{q_A=0}^{q_\mathrm{total}}\left(\binom{q_A+N_A-1}{q_A}\binom{q_B+N_B-1}{q_B}\right)\right)x^{q_\mathrm{total}}
\end{equation}
\end{strip}

Text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text.

\lipsum[3-5]
\end{document} 

Resizebox should work.

\documentclass[twocolumn]{ltxdoc}
\usepackage{amsmath}
\usepackage{lipsum}
\usepackage{graphics}
\begin{document}
\lipsum[1]
\begin{equation}
\resizebox{0.5\textwidth}{!}{$\sum_{q_\mathrm{total}=0}^{q_A+N_A+q_B+N_B-2}\left(\sum_{q_A=0}^{q_\mathrm{total}}\left(\binom{q_A+N_A-1}{q_A}\binom{q_B+N_B-1}{q_B}\right)\right)x^{q_\mathrm{total}}$}
\end{equation}
\lipsum[2-4]
\end{document}

Or maybe consider splitting your equation.

I don't think you have a much better option than this. Btw, you seem to use q_A for two different things.

\documentclass[twocolumn]{ltxdoc}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\sum_{q_\mathrm{tot}=0}^{Q_A+Q_B-2}\sum_{q_A=0}^{q_\mathrm{tot}}{\textstyle\binom{Q_A-1}{q_A}\binom{Q_B-1}{q_B}}x^{q_\mathrm{tot}},
\end{equation}
where $Q_A=q_A+N_A$ and $Q_B=q_B+N_B$.
\end{document}