In this post we evaluate the sum . Then we’ll generalize it and evaluate .

The key tools we need for the first sum are the *trinomial revision* identity, , and the *parallel summation *identity, . Using trinomial revision, we have

.

We can evaluate the remaining sum using parallel summation, but we need to pull a variable switch first. Replace *k* with and then apply parallel summation to obtain

.

We thus have the identity

where the last step follows thanks to some algebra simplification with factorials.

To evaluate the second sum we’ll need an identity that’s like an upper-index version of Vandermonde’s convolution: . Otherwise, the derivation for the second sum proceeds just as the first one does up through the point where we replace *k* with . Continuing from there and applying the upper Vandermonde identity, we have

We can rewrite this and perform some more algebra simplification to get our generalization:

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