Recall that $(a;q)_0:=1,\,(a;q)_n=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ and $(a;q)_{\infty}=(1-a)(1-aq)(1-aq^2)\cdots$. Let's introduce the following (generalized) concept.

A *colored overpartition* (COP) is a partition where the last occurrence of each distinct number may receive any one of $c$ colors. The number of such partitions of $n$ we denote by $\overline{p}_c(n)$. For example, the $8$ COP partitions of $n=3$ with $c=2$ colors are:
$$3,\, \overline{3}, \, 2+1, \, \overline{2}+1, \, 2+\overline{1}, \, \overline{2}+\overline{1}, \, 1+1+1, \, 1+1+\overline{1}.$$
**Note.** In the literature, (i) ordinary partitions $p(n)=\overline{p}_1(n)$; (ii) overpartitions $\overline{p}(n)=\overline{p}_2(n)$.

Question.Does this generating function hold true? $$\sum_{n\geq0}\overline{p}_c(n)q^n=\frac{((1-c)q;q)_{\infty}}{(q;q)_{\infty}}.$$

`$COP$`

uses mathemarics spacing for what is an acronym, so looks wrong. I think the small-caps`\textsc{COP}`

, italic`\textit{COP}`

or just plain`COP`

would look better. $\endgroup$ – J.J. Green Mar 7 '17 at 12:46