Don't forget that HP was one of the printer companies that used all the ink colors (black, blue, magenta, and yellow) when printing a back and white document. They got their hand slapped for that stunt. They claimed it gave darker black lettering.

@Derlin
I don't think that the printer does any different motions when using an HP cartridge versus a third party unit. The only thing I can think of is that the print head, which is part of the ink cartridge, may clog