Evaluate
Source: ejcnt
First, we may write:
The first equality comes from the fact that when and and the second because: (they count complementary combinations)
Now consider the Ordinary Generating Function :
A bit of Snake-Oil ( Exchange the summation order):
The inner series is: ( This can be obtained differentiating repeatedly )
So:
Now we read off the coefficients of the OGF bearing in mind that to obtain: