How often can a number occur in Pascals Triangle?

Question

Each number occurs at least twice as $a={a\choose1}={a\choose a-1}$. If a number occurs somewhere else in the triangle (most likely twice, if it's not of the form ${2a\choose a}$) then that number occurs $4$ times. After that, it becomes interesting. I found the following with a simple script:

$$120={120\choose1}={16\choose 2}={10\choose3}={10\choose7}={16\choose 14}={120\choose119}$$

$$210={210\choose1}={21\choose 2}={10\choose4}={10\choose6}={21\choose 19}={210\choose209}$$

$$1540={1540\choose1}={56\choose 2}={22\choose3}={22\choose19}={56\choose 54}={1540\choose1539}$$

$$7140={7140\choose1}={120\choose 2}={36\choose3}={36\choose33}={120\choose 118}={7140\choose7139}$$

And a special one, that did not only occur six times, but eight:

$$3003={3003\choose1}={78\choose 2}={15\choose5}={14\choose6}={14\choose8}={15\choose10}={78\choose 76}={3003\choose3002}$$

I only checked the numbers up to $10000$; so here's my question

Besides $1$, are there other numbers that occur infinitely often? Is there an upper bound known to how many times a number can occur in Pascal's Triangle?

Searching on OEIS with just 120, 210, 1540 from your own data, already gives some references. — Jeppe Stig Nielsen, Apr 02 '17 at 22:21
"Besides 11, are there other numbers that occur infinitely often?" - no, and you can confirm this by noting that after row $n$ ($n>1$), the number $n$ cannot appear. — Glen O, Apr 03 '17 at 02:36
Related MO post: Singmaster's conjecture. I found the link in answers to this question: Are there surprisingly identical binomial coefficients? — Martin Sleziak, Apr 03 '17 at 16:29
If a number occurs somewhere else in the triangle not of the form (2aa)) then that number occurs 4 times. Number 6 occured 3 times. — Tomilov Anatoliy, Feb 11 '18 at 19:55
You misread that. If it occurs again, and it's not of the form ${2a\choose a}$, then it occurs (at least) $4$ times. $6$ is ${2\cdot 2\choose 2}$, so is of that form. — , Feb 13 '18 at 09:13

score 52 · Accepted Answer · answered Apr 02 '17 at 18:24

52

Singmaster's conjecture (Wikipedia article) is that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than 1). The article has all of the latest developments in this direction, and as explained in the article,

It is clear that the only number that appears infinitely many times in Pascal's triangle is $1$, because any other number $x$ can appear only within the first $x + 1$ rows of the triangle.

answered Apr 02 '17 at 18:24

Zev Chonoles

132,937

10

I feel silly for asking whether numbers can occur infinitely often now, that was pretty obvious. Arbitrarily many times, however, remains interesting. Thanks! – Apr 02 '17 at 18:29

How often can a number occur in Pascals Triangle?

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