Smallest number to appear n times in Pascal's triangle

David C Terr David_C_Terr at
Mon Nov 22 19:52:45 CET 2004

Interesting! Do you know if the number of times a binomial coefficient 
greater than 1 appears is bounded? If so, what's an upper bound? By the 
way, it seems that numbers which appear an odd number of times should be 
rarer since Pascal's triangle is symmetric, so at least one of the 
coefficients would need to have the form (2n n).


"Paul D. Hanna" <pauldhanna at>
11/20/2004 12:56 PM

        To:     seqfan at
        Subject:        Re: Smallest number to appear n times in Pascal's triangle

If there are no binomial coefficients that appear exactly 5 times, 
then it may be best to change the name of A003015 to be 
"Numbers that occur 6 or more times in Pascal's triangle.": 
Sequence:  1,120,210,1540,3003,7140,11628,24310, 
Name:      Numbers that occur 5 or more times in Pascal's triangle. 
On Fri, 19 Nov 2004 23:14:27 -0800 David Wasserman <dwasserm at> writes:
> I didn't see the message from Isabel Lugo.  Do we know if there's a 
> number that appears exactly 5 times?
> >Because of the imprecise way I worded it, Hugo Pfoertner is 
> absolutely
> >right to answer A062527, where a(5)=a(6). If I had instead said
> >"Smallest number to appear exactly n times in Pascal's triangle", 
> then
> >it'd be different. To look for that fifth term where Isabel Lugo 
> said,
> >at 61218182743304701891431482520, appears to be quite a formidable
> >task. (Thanks for the MathWorld link, by the way, I hadn't thought 
> to
> >look there).
> >
> >Alonso

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