It's not too difficult to see that (and understand why) in a base n system the ciphers of $(11_n)^k$ are equivalent to the k-th (0-indexed) row of pascals triangle until one of the numbers becomes greater than or equal to the base and thereby changing number represented by the next cipher.
Would something like $(11_\infty)^i$ or $\lim\limits_{n \to \infty} (11_n)^k$
- make sense at all?
- indeed represent the k-th row of pascals triangle?