Smallest number to appear n times in Pascal's triangle

[Henry in Rotherhithe]

>  > Do you know if the number of times a binomial coefficient
> greater than 1
> appears is bounded?
>
> I may be mistaken, but I think so.  choose(n,k) grows like O(n^k),
> right?  So they will eventually exceed any given C.  The first to go will
> be the central column, the last to sail over the horizon will be the pair
> at C=choose(C,1)=choose(C,C-1).  So there's a bounded region
> which contains
> all instances of C.

I think you are answering a different question.  As you say, any particular
number >1 cannot appear an infinite number of times, but the question is
whether there is an overall limit applying to all numbers.  There is not for
the triangle n^min(k,n-k).

Henry Bottomley