Evaluate

Source: ejcnt

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- Apr 4th 2009, 11:12 AMNonCommAlgBinomial coefficiens: an identity
Evaluate

Source: ejcnt - Apr 13th 2009, 05:46 PMPaulRS
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: - Apr 13th 2009, 06:04 PMNonCommAlg