Pascal Triangle

I suppose we can all dream the Pascal Triangle.
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
etc.
Until today I did not know that we could simply read the following sum from the triangle.
$$ s(n) = \sum_{i=1}^{n} i \cdot ( n-i ) $$
Like for example 100 * 1 + 100 * 2 + 100 * 3 + 100 * 4 + ... + 1 * 100 = ? The answer is in the triangle. Amazing. Or perhaps "fascinating" is a better word.