Is there a word like cardinal or ordinal but for the “single, double, triple” series?

These are called multiplicatives by Fowler (1850). The term multiplicative is also seen to describe the words once, twice, thrice, and it seems reasonable to distinguish the former as multiplicative adjectives, and the latter as multiplicative adverbs.

There doesn't seem to be.

Wikipedia says they are called "tuples" and associates them with ordered sets, but this language is specific to mathematics.

http://en.wikipedia.org/wiki/Tuple

There is no definition of "tuple" in Merriam-Webster, though there is a definition for the noun combining form "-tuple." It is the "set of (so many) elements —usually used of sets with ordered elements ."

There is no explanation in the Chicago Manual of Style, though rule 12.29 refers to the "ordered n-tuple of objects," which seems again to be a particular, technical usage.

I'm pretty sure Dave is right; there does not seem to be a term or name for this sequence.

So why not have a little fun inventing one? If cardinal refers to power or amount (one, two, three), and if ordinal refers to position (first, second, third), then single, double, triple would be referring to a mathematical relationship. We might invent a term such as multipal, but that would be confused with multiple. And multiplal would be terribly awkward. So how about relational? The relational numbers. Doesn't sound too bad. But, hey, it's just a suggestion.