1001, 2002, 3003, ... featuring in Pascal's triangle

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

David W. Wilson reported that the most prevalent number in Pascal's triangle
(other than 1) is 3003, which occurs 8 times :

3003 = (14 6) = (14 8) = (15 5) = (15 10) =
(78 2) = (78 76) = (3003 1) = (3003 3002)

No other number is known to occur a larger finite number of times.