Mathematica: 36 (41?)
Mathematica has the `Binomial` function, but that takes the fun out of this. I propose:
NestList[{0,##}+{##,0}&@@#&,{1},n-1] The line above will render a ragged array such as:
{{1}, {1, 1}, {1, 2, 1}, {1, 3, 3, 1}, {1, 4, 6, 4, 1}, {1, 5, 10, 10, 5, 1}, {1, 6, 15, 20, 15, 6, 1}} Since this is a basic format in Mathematica I thought it would be acceptable, but as I read the rules again, I think it may not be. Adding Grid@ will produce unequivocally acceptable output, for a total of 41 characters:
Grid@NestList[{0,##}+{##,0}&@@#&,{1},n-1] n = 6:
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1