I will start with extended chain notation. For more extension, I will write numbers below the chain.

This looks like chemical equation, so I call {what is below the chain} {catalyst}. The catalyst makes number really bigger.

Catalyst must not be one. Some reactions require more than one catalysts.

(b stories)

Next, using the same rule, we can define up to .

Then,

(b stories)

Keep going on until . At this stage, I think this is as big as .

Next, three catalysts come.

(b stories)

See the pattern? As you know, definition can be up to n-entry.

This notation makes up to .

If you have a question, ask in a comment.

For comment, I will write generalized formula.

If the last number under the chain is not zero, it is same as extended chain (so you can ignore what is the left).

(b stories)