I deal with a lot of double sums / double products. I would like to be able to massage TeX into producing something like this:

(that was made with the gimp) ... and a product in product symbol as well.

up vote 7 down vote accepted

You can have a look at the section Joining and overlapping existing symbols in manual of Comprehensive Symbol List.

\documentclass{article}

\begin{document}

\newcommand\SUM{\mathop{%
  \ooalign{%
    \hfil$\displaystyle\sum$\cr
    \hfil$\textstyle\sum$\cr}}}

\[
  \SUM A_i
\]

\end{document}

enter image description here

For what it is worth, Unicode defines

⅀ U+2140 DOUBLE-STRUCK N-ARY SUMMATION

If you are able to use unicode-math (with XeLaTeX or LuaLaTeX) and appropriate math fonts, you can access that symbol with \mathbb{\sum}.

STIX/XITS Math gives stix double-struck sum and Cambria Math gives cambria double-struck sum.

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