# Can Pascal's Triangle be expressed as a exponential equation?

Pascal's triangle seems to follow a pattern. 11row-1 outputs PascalTriangle(row) while row ≤ 5, but 116-1 outputs 161051. However if you used Base100, you could go further.

So, my conjecture is that 11row-1 will be equal to the corresponding row in Pascal's Triangle, given that all calculations are done without a Base. For example, 116-1 would return 15101051, not 161051.

I asked my teacher about this, and we argued about it, him saying that it would have to break at some point, but I argued that it couldn't, it had no base, so there would never be carrying over, and thus no discrepancies.

• Base 0? That doesn't make any sense. – lordoftheshadows Nov 30 '16 at 4:14
• The pattern you've discovered is a known (perhaps not widely known) property of Pascal's triangle. Recall that each row gives the coefficients of the expansion of $(1+x)^n$ and that any number can be written as a sum of powers of 10. Substitute $x=10$ and the pattern should make sense for the first five rows. – Corellian Nov 30 '16 at 4:24
• @Brody, thanks for your answer. I can't seem to upvote it. Don't know why. – Aanand Kainth Nov 30 '16 at 4:43
• I don't think you have enough reputation yet for that privilege. Anyway, it's just a comment on your post, not an actual answer. A good answer might address how (if even possible) to have each row truly look like the digits of $11^n$ in some base, or something vaguely of that effect. – Corellian Nov 30 '16 at 4:57