How to adjust self defined multiple product signs

Question

The latex code below produces multiple product signs:

\documentclass{article}
\usepackage{amssymb,amsmath,graphicx,color}

\newcommand{\invcircledast}{%
\mathbin{\vphantom{\circledast}\text{%
\ooalign{\smash{\blackcircle}\cr
         \hidewidth\smash{\textcolor{white}{$*$}}\hidewidth\cr
        }%
}}%
}
\newcommand{\blackcircle}{\raisebox{-.4ex}{\scalebox{1.66}{$\bullet$}}}

\makeatletter
\newcommand\bigCircledast{\mathop{\mathpalette\b@gCircledast\relax}}
\newcommand\b@gCircledast[2]{%
\vcenter{\hbox{\m@th
\scalebox{\ifx#1\displaystyle 2.3\else1.2\fi}{$#1\circledast$}%
}}%
}
\newcommand\bigInvcircledast{\mathop{\mathpalette\b@gInvcircledast\relax}}
\newcommand\b@gInvcircledast[2]{%
\vcenter{\hbox{\m@th
\scalebox{\ifx#1\displaystyle 2.3\else1.2\fi}{$#1\invcircledast$}%
}}%
}
\makeatother

\begin{document}
\[
\bigoplus_{i=1}^n % just for comparison
\bigCircledast_{i=1}^n % just for comparison
\bigInvcircledast_{i=1}^n (a_i\invcircledast b_i)
\]
\end{document}

My question now is how to adjust the product signs \bigCircledast and \bigInvcircledast such that the disks have the same size as \bigoplus AND that the sub- and superscripts are on the same line as the sub- or the superscripts of \bigoplus respectively.

Compare also the questions and answers of Opposite product signs in the sense that black and white are interchanged and Filled \Diamond-product sign.

Most of the ideas used in the code above came from egreg. See also the links above.

Answer 1

Here is one solution to this question: Essentially I added \vcenter{\hbox{\rule{0pt}{16pt}}} at the end of the command definitions. The disadvantige of this solution is its dependence on the font size. For example, if one uses \Huge to enlarge the font size, the height of \rule needs to be adjusted to 40pt.

The whole code of this solution is:

\documentclass{article}
\usepackage{amssymb,amsmath,graphicx,color}

\newcommand{\invcircledast}{%
\mathbin{\vphantom{\circledast}\text{%
\ooalign{\smash{\blackcircle}\cr
         \hidewidth\smash{\textcolor{white}{$*$}}\hidewidth\cr
        }%
}}%
}
\newcommand{\blackcircle}{\raisebox{-.4ex}{\scalebox{1.66}{$\bullet$}}}

\makeatletter
\newcommand\bigCircledast{\mathop{\mathpalette\b@gCircledast\relax}}
\newcommand\b@gCircledast[2]{%
\vcenter{\hbox{\m@th
\scalebox{\ifx#1\displaystyle 2.1\else1.2\fi}{$#1\circledast$}%
}}\vcenter{\hbox{\rule{0pt}{16pt}}}%
}
\newcommand\bigInvcircledast{\mathop{\mathpalette\b@gInvcircledast\relax}}
\newcommand\b@gInvcircledast[2]{%
\vcenter{\hbox{\m@th
\scalebox{\ifx#1\displaystyle 2.1\else1.2\fi}{$#1\invcircledast$}%
}}\vcenter{\hbox{\rule{0pt}{16pt}}}%
}
\makeatother

\begin{document}%\Huge
\[
\bigoplus_{i=1}^n % just for comparison
\bigCircledast_{i=1}^n
\bigInvcircledast_{i=1}^n (a_i\invcircledast b_i)
\]
\end{document}